Chapter 1 Introduction to Measurement Systems

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Advanced Measurement Systems and Sensors Dr. Ibrahim Al-Naimi

Chapter one

Introduction to Measurement Systems

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Outlines

• Control and measurement systems • Transducer/sensor definition and classifications • Signal conditioning definition and classifications • Units of measurements • Types of errors • Transducer/sensor transfer function • Transducer characteristics • Statistical analysis

Closed-loop Control System

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Measurement System

Transducer and Signal Conditioning

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Transducer Element

• The Transducer is defined as a device, which when actuated by one form of energy, is capable of converting it to another form of energy. The transduction may be from mechanical, electrical, or optical to any other related form. • The term transducer is used to describe any item which changes information from one form to another.

Transducer and Sensor • Transducers are elements that respond to changes in the physical condition of a system and deliver output signals related to the measured, but of a different form and nature.

• Sensor is the initial stage in any transducer.

• The property of transducer element is affected by the variation of the external physical variable according to unique relationship.

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Transducers classification

• Based on power type classification - Active transducer (Diaphragms, Bourdon Tubes, tachometers, piezoelectric, etc…) - Passive transducer (Capacitive, inductive, photo, LVDT, etc…)

Transducers classification

• Based on the type of output signal - Analogue Transducers (stain gauges, LVDT, etc…) - Digital Transducers (Absolute and incremental encoders)

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Transducers classification

• Based on the electrical phenomenon or parameter that may be changed due to the whole process. - Resistive transducer - Capacitive transducer - Inductive transducer - Photoelectric transducer

Transducers and Sensors

According to the physical variable to be measured, the transducers can be classified to: • Mechanical Transducers • Thermal Transducers • Optical Transducers

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Transducer Element

• Single stage (direct) • Double stage (indirect)

Single Stage Transducer

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Double Stage Transducer

Typical Examples of Transducer Elements

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Typical Examples of Transducer Elements

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Signal Conditioning

Signal conditioning refers to operations performed on signals to convert them to a form suitable for interfacing with other elements in control system • Analog signal conditioning. • Digital signal conditioning.

Analog Signal Conditioning

• Analog signal conditioning provides the operations necessary to transform a transducer output into a form necessary to interface with other elements in control system.

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Analog Signal Conditioning

• It is possible to categorize signal conditioning into several general types: - Signal level and bias changes - Linearization - Conversion - Filtering

Unites of Measurements

• The base units, which have been established in The 14th General Conference on Weights and Measures (1971), will be used in this course. • The base measurement system is known as SI which stands for French “Le Syste´me International d’Unite´s”.

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Unites of Measurements

Block Diagram Symbols

• A block diagram approach is very useful in discussing the properties of elements and systems.

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Type of Errors • Systematic errors: Errors that tend to have the same magnitude and sign for the given set of conditions, such as instrument, environmental, and loading errors. This type of error can be eliminated by applying the proper calibration.

• Random (or Accidental) errors: These errors are of variable magnitude and sign and do not obey any known law. Theses errors caused due to random variation in the parameter or the system of measurement, such as the accuracy of measuring small quantities. This type of error can be calculated by using statistical analysis.

Type of Errors • Gross errors: These are generally the fault of the person using the instruments, and are due to: – Incorrect reading of instruments . – Incorrect recording of experiment data.. – Incorrect use of instruments.

These errors may be of any magnitude and cannot be subjected to mathematical treatment.

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Statistical Analysis of Error in Measurement • Mean Value x  x ......  x x x  1 2 n   n n n • Standard Deviation

d 2  d 2 ......  d 2   1 2 n Where: n

d1  x1  x

d2  x2  x Note: if the number of samples is less than 20, (n) in the standard deviation law will be replaced by (n-1)

Statistical Analysis of Error in Measurement Interpretation of Standard deviation • If the error is truly random and we have large sample of readings, the data and the standard deviation is related to the normal probability curve. • The normal probability curve has the following characteristics: – 68% of readings lie within ±1σ of the mean – 95.5% of readings lie within ±2σ of the mean – 99.7% of readings lie within ±3σ of the mean

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Transfer Function/Mathematical Model

• An ideal or theoretical input–output (stimulus– response) relationship exists for every sensor. • This input–output relationship may be expressed in the form of a table of values, graph, mathematical formula, or as a solution of a mathematical equation. • If the input–output function is time invariant (does not change with time) it is commonly called a static transfer function or simply transfer function.

Transfer Function/Mathematical Model • A static transfer function represents a relation between the input stimulus s and the electrical signal E produced by the sensor at its output. This relation can be written as E=f(s). • Normally, stimulus s is unknown while the output signal E is measured and thus becomes known. The value of E that becomes known during measurement is a number (voltage, current, etc.) that represents stimulus s. • A job of the designer is to make that representation as close as possible to the true value of stimulus s.

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Transfer Function/Mathematical Model • In reality, any sensor is attached to a measuring system. One of the functions of the system is to “break the code E” and infer the unknown value of s from the measured value of E. Thus, the measurement system shall employ an inverse transfer function s=f -1(E)=F(E), to obtain (compute) value of the stimulus s. • It is usually desirable to determine a transfer function not just of a sensor alone, but rather of a system comprising the sensor and its interface circuit.

Transfer Function/Mathematical Model • Preferably, a physical or chemical law that forms a basis for the sensor’s operation should be known. • If such a law can be expressed in form of a mathematical formula, often it can be used for calculating the sensor’s inverse transfer function by inverting the formula and computing the unknown value of s from the measured output E. • In practice, readily solvable formulas for many transfer functions, especially for complex sensors, does not exist and one has to resort to various approximations of the direct and inverse transfer functions.

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Function Approximations • Approximation is a selection of a suitable mathematical expression that can fit the experimental data as close as possible. • The act of approximation can be seen as a curve fitting of the experimentally observed values into the approximating function. • The approximating function should be simple enough for ease of computation and inversion and other mathematical treatments, for example, for computing a derivative to find the sensor’s sensitivity.

Function Approximations • Initially, one should check if one of the basic functions can fit the data and if not, then resort to a more general curve-fitting technique, such as a polynomial approximation. • The simplest model of a transfer function is linear. It is described by the following equation:

• As shown in the following figure, it is represented by a straight line with the intercept A, which is the output signal E at zero input signal (s=0). The slope of the line is B (sensitivity).

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Function Approximations

• As shown in the following figure, it is represented by a straight line with the intercept A, which is the output signal E at zero input signal (s=0). The slope of the line is B (sensitivity).

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Function Approximations • Note that the previous equation assumes that the transfer function passes, at least theoretically, through zero value of the input stimulus s. • In many practical cases it is just difficult or impossible to test a sensor at a zero input. Thus, in many linear or quasilinear sensors it may be desirable to reference the sensor not to the zero input but rather to some more

practical input reference value s0. If the sensor response is E0 for some known input stimulus s0, the previous equation can be rewritten in a more practical form:

Function Approximations

• The reference point has coordinates s0 and E0. For a particular case where s0=0, Eq. (2) becomes Eq. (1) and E0=A. • The inverse linear transfer function for computing the input stimulus from the output E is given by:

• Note that three constants shall be known for computing

the stimulus s, sensitivity B, s0 and E0 of the reference point.

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Function Approximations • Very few sensors are truly linear. • In the real world, at least a small nonlinearity is almost always present, especially for a broad input range of the stimuli. Thus, the previous equations represent just a linear approximation of a nonlinear sensor’s response, where a nonlinearity can be ignored for the practical purposes. • In many cases, when nonlinearity cannot be ignored, the transfer function still may be approximated by a group of linear functions (linear piecewise approximation).

Function Approximations • A nonlinear transfer function can be approximated by a nonlinear mathematical function, such as, logarithmic approximation function, exponential function, and power function. • All the above three nonlinear approximations possess a small number of parameters that shall be determined during calibration. A small number of parameters makes them rather convenient, provided that they can fit response of a particular sensor. It is always useful to have as small a number of parameters as possible, not the least for the sake of lowering cost of the sensor calibration. The fewer parameters, the smaller the number of the measurements to be made during calibration.

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Function Approximations The logarithmic approximation function

Function Approximations The exponential function

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Function Approximations The power function

Linear Regression • If measurements of the input stimuli during calibration cannot be made consistently with high accuracy and large random errors are expected, the minimal number of measurements will not yield a sufficient accuracy. • To cope with random errors in the calibration process, a method of least squares could be employed to find the slope and intercept. • In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables.

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Linear Regression • More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. • Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed. • a function of the independent variables called the regression function is to be estimated.

Linear Regression • In regression analysis, it is also of interest to characterize the variation of the dependent variable around the prediction of the regression function using a probability distribution. • Regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. • Many techniques for carrying out regression analysis have been developed. Familiar methods such as linear regression and least squares regression are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data.

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Linear Regression • The least squares method is a form of mathematical regression analysis that finds the line of best fit for a set of data, providing a visual demonstration of the relationship between the data points. Each point of data is representative of the relationship between a known independent variable and an unknown dependent variable. • The least squares method is a statistical procedure to find the best fit for a set of data points by minimizing the sum of the offsets or residuals of points from the plotted curve. • Least squares regression is used to predict the behavior of dependent variables.

Linear Regression • The procedure to find the linear approximation function using least square method is as follows: - Measure multiple (k) output values E at the input values s over a substantially broad range, preferably over the entire sensor span. - Use the following formulas for a linear regression to determine intercept A and slope B of the best-fitting straight line equation (i.e. E=A+Bs)

• Where Σ is the summation over all k measurements. When the constants A and B are found, the result equation can be used as a linear approximation of the experimental transfer function.

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Polynomial Approximations • A sensor may have such a transfer function that none of the above basic functional approximations would fit sufficiently well. Thus, several old and reliable techniques can be used. One is a polynomial approximation, that is, a power series. • Any continuous function, regardless of its shape, can be approximated by a power series. • For example, the exponential function of can be approximately calculated from a third-order polynomial by dropping all the higher terms of its series expansion

Polynomial Approximations • In many cases, it is sufficient to see if the sensor’s response can be approximated by the second or third degree polynomials to fits well enough into the experimental data. • These approximation functions can be expressed respectively as

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Polynomial Approximations • The factors a and b in the previous equations are the constants that allow shaping the curves into a great variety of the practical transfer functions. • The inverse transfer function can be approximated by a second or third degree polynomial:

• In some cases, especially when more accuracy is required, the higher order polynomials should be considered because the higher the order of a polynomial the better the fit.

Linear Piecewise Approximation • A linear piecewise approximation is a powerful method to employ in a computerized data acquisition system. • The idea behind it is to break up a nonlinear transfer function of any shape into sections and consider each such section being linear as described by the previously mentioned linear equations. • Curved segments between the sample points (knots) demarcating the sections are replaced with straight-line segments, thus greatly simplifying behavior of the function between the knots. In other words, the knots are graphically connected by straight lines. This can also be seen as a polygonal approximation of the original nonlinear function

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Linear Piecewise Approximation

Linear Piecewise Approximation • The previous figure shows the linear piecewise approximation

of a nonlinear function with the knots at input values s0, s1, s2, s3, s4, and the corresponding output values n0, n1, n2, n3, n4. • The error of a piecewise approximation can be characterized by the maximum deviation δ of the approximation line from the real curve. Different definitions exist for this maximum deviation (mean square, absolute max, average, etc.); but whatever is the adopted metric, the larger δ calls for a greater number of samples, that is a larger number of sections with the idea of making this maximum deviation acceptably small. In other words, the larger the number of the knots the smaller the error.

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Spline Interpolation • interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate, i.e., estimate the value of that function for an intermediate value of the independent variable.

Spline Interpolation • Approximations by higher order polynomials (third order and higher) have some disadvantages; the selected points at one side of the curve make strong influence on the remote parts of the curve. • This deficiency is resolved by the spline method of approximation. • In a similar way to a linear piecewise interpolation, the spline method is using different third-order polynomial interpolations between the selected experimental points called knots. • The spline interpolation can utilize polynomials of different degrees, yet the most popular being cubic (third order) polynomials.

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Interpolation Vs. Regression • Interpolation is a method of constructing new data points within the range of a discrete set of known data points. • It is often required to interpolate the value of that function for an intermediate value of independent variable. This can be achieved by curve fitting or regression analysis. • There are many different interpolation methods. Some examples of this include piecewise constant interpolation, linear interpolation, polynomial interpolation, spline interpolation and Gaussian processes.

Interpolation Vs. Regression

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Multidimensional Transfer Functions • A sensor transfer function may depend on more than one input variable. That is, the sensor’s output may be a function of several stimuli. • One example is the transfer function of a thermal radiation (infrared) sensor. This function has two

arguments—two temperatures: Tb, the absolute temperature of an object of measurement and Ts, the absolute temperature of the sensing element. Thus, the sensor’s output voltage V is proportional to a difference of the fourth-order parabolas:

Multidimensional Transfer Functions • The graphical representation of a two-dimensional transfer function of in the previous equation is shown in the figure.

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Transducer Characteristics

• Static characteristics. • Dynamic Characteristics (self study).

Static Characteristics

• Accuracy Accuracy is defined as the closeness of the transducer output to the true value of the measured quantity. However, in usual practice, it is specified as the inaccuracy of measurement from the true value by calculating the absolute and percent error. The accuracy of an instrument depends on the various systematic errors.

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Static Characteristics • Accuracy

- Absolute Error eA  Yn  X n

Where: Yn= expected value

Xn= measured value (could be μ if the device is suffering from random and systematic errors)

- Percent Error Yn  X n eP  100% Yn

Static Characteristics Inaccuracy/Measurement Uncertainty • Inaccuracy or measurement uncertainty is the extent to which a reading might be wrong and is often quoted as a percentage of the full-scale (f.s.) reading of an instrument. • Because the maximum measurement error in an instrument (measurement system) is usually related to the full-scale reading of the instrument, measuring quantities that are substantially less than the full-scale reading means that the possible measurement error is amplified.

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Static Characteristics Inaccuracy/Measurement Uncertainty • For this reason, it is an important system design rule that instruments are chosen such that their range is appropriate to the spread of values being measured in order that the best possible accuracy is maintained in instrument readings. • Clearly, if we are measuring pressures with expected values between 0 and 1 bar, we would not use an instrument with a measurement range of 0–10 bar.

Static Characteristics Inaccuracy/Measurement Uncertainty Example:

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Static Characteristics Tolerance • Tolerance is a term that is closely related to accuracy and defines the maximum error that is to be expected in some value. • While it is not, strictly speaking, a static characteristic of measuring instruments, it is mentioned here because the accuracy of some instruments is sometimes quoted as a tolerance value. • When used correctly, tolerance describes the maximum deviation of a manufactured component from some specified value.

Static Characteristics

• Precision/Repeatability: Precision is defined as the ability of the instrument to reproduce a certain set of readings within a given accuracy. Repeatability is defined as the ability of the instrument to reproduce a group of measurements of the same measured quantity, made by the same observer, using the same instrument, and under the same conditions. The precision of the instrument depends on the factors that cause random errors.

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Static Characteristics

• Precision

2 precision%  100 Fullscale(output)

Static Characteristics

• Accuracy Versus Precision

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Static Characteristics • Resolution or Discrimination: It is defined as the smallest increment in the measured value that can be detected with certainty by the instrument. • Resolution is also defined as the largest change in input that can occur without any corresponding change in output. Resolution is normally expressed as a percentage of input full scale as: I resolution%  100 Fullscale

Static Characteristics • Range or Span The range or span of an instrument defines the minimum and maximum values of a quantity that the instrument is designed to measure.

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Static Characteristics

Static Sensitivity (scale factor or gain): • The static sensitivity of measurement is a measure of the change in instrument output that occurs when the quantity being measured changes by a given amount. q Slop  K  o qi Where q0 and qi are the values of output and input signals respectively.

Static Characteristics - If the relationship between input and output is linear, the sensitivity is constant - If the relationship between input and output is nonlinear, the sensitivity varies with the input value

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Static Characteristics

Linearity/Nonlinearity: It is the closeness to which a curve approximates a straight line. The linearity is one of the most desirable features of any instrument. However, linearity is not completely achieved.

Static Characteristics • Many instruments are designed to achieve a linear relationship between the applied static input and indicated output values. Such a linear static calibration curve would have the general form

where the curve fit yL(x) provides a predicted output value based on a linear relation between x and y. • However, in real systems, truly linear behavior is only approximately achieved. As a result, measurement device specifications usually provide a statement as to the expected linearity of the static calibration curve for the device.

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Static Characteristics

• The relationship between yL(x) and measured value y(x) is a measure of the nonlinear behavior of a system:

• where uL(x) is a measure of the linearity error that arises in describing the actual system behavior by the equation. Such behavior is illustrated in the following figure in which a linear curve has been fit through a calibration dataset.

Static Characteristics Linearity/Nonlinearity:

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Static Characteristics Linearity/Nonlinearity: • For a measurement system that is essentially linear in behavior, the extent of possible nonlinearity in a measurement device is often specified in terms of the maximum expected linearity error as a percentage of full-scale output range.

Static Characteristics Dead Zone (Dead Band): Is the largest change of the measured quantity in which the instrument does not respond.

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Static Characteristics

Drift (sensitivity to disturbance): • All calibrations and specifications of an instrument are only valid under controlled conditions of temperature, pressure, and so on. These standard ambient conditions are usually defined in the instrument specification (datasheet). • As variations occur in the ambient temperature, certain static instrument characteristics change, and the sensitivity to disturbance is a measure of the magnitude of this change.

Static Characteristics

Drift (sensitivity to disturbance): • Such environmental changes affect instruments in two main ways, known as zero drift and sensitivity drift. Zero drift is sometimes known by the alternative term, bias. • Zero drift or bias describes the effect where the zero reading of an instrument is modified by a change in ambient conditions. This causes a constant error that exists over the full range of measurement of the instrument. • Zero drift is normally removable by calibration.

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Static Characteristics

Drift (sensitivity to disturbance): • The typical unit by which such zero drift is measured is volts/oC. This is often called the zero drift coefficient related to temperature changes

(KI). • If the characteristic of an instrument is sensitive to several environmental parameters, then it will have several zero drift coefficients, one for each environmental parameter.

Static Characteristics Drift (sensitivity to disturbance): • Sensitivity drift (also known as scale factor drift) defines the amount by which an instrument’s sensitivity of measurement varies as ambient conditions change. • It is quantified by sensitivity drift coefficients that define how much drift there is for a unit change in each environmental parameter that the instrument characteristics are sensitive to (KM). • Sensitivity drift is measured in units of the form (angular degree/bar)/oC. • Some instruments may suffer from both type of drifts.

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Static Characteristics

Drift (sensitivity to disturbance):

Static Characteristics

Drift (sensitivity to disturbance):

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Static Characteristics Hysteresis: It is defined as the magnitude of error caused in the output for a given value of input, when this value is approached from opposite direction, i.e. from ascending order and then descending order. This is caused by backlash, elastic deformation, magnetic characteristics, and frictional effect.

Static Characteristics Hysteresis: Hysteresis is usually specified for a measurement system in terms of the maximum hysteresis error as a percentage of full-scale output range.

Hysteresis Hysteresis %  MAX 100 MAX Fullscale

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Static Characteristics Hysteresis:

Static Characteristics Overall Instrument Error and Uncertainty : • An estimate of the overall instrument error is made by combining the estimates of all known errors into a term called the instrument uncertainty. The estimate is computed from the square root of the sum of the squares (SRSS) of all known uncertainty values. For M known

errors, the overall instrument uncertainty, uc, is estimated by: • For example, for an instrument having known hysteresis, linearity, and sensitivity drift errors, the instrument uncertainty is estimated by

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Loading Effect (Impedance Load) • Any measuring instrument with an input signal source would extract some energy, thereby changing the value of the measured variable. This means that the input signal suffers a change by virtue of the fact that it is being measured. This effect is termed loading. • One of the most important concerns in analog signal conditioning is the loading of one circuit by another. This introduces uncertainty in the amplitude of a voltage as it passed through the measurement process. If this voltage represents some process variable, then we have uncertainty in the value of the variable. • The loading error (effect) in the measurement system can never be zero, but it must be made as small as possible. How????

Loading Effect (Impedance Load) • Thévenin’s theorem states that any network consisting of linear impedances and voltage sources can be replaced by an equivalent circuit consisting of

a voltage source ETh and a series impedance ZTh. The source ETh is equal to the open circuit voltage of the network across the output terminals, and ZTh is the impedance looking back into these terminals with all voltage sources reduced to zero and replaced by their

internal impedances. Thus connecting a load ZL across the output terminals of the network is equivalent to

connecting ZL across the Thévenin circuit.

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Loading Effect (Impedance Load)

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Loading Effect (Impedance Load)

• Example: An amplifier outputs a voltage that is ten times the voltage on its input terminals. It has an input resistance of 10 kΩ. A sensor outputs a voltage proportional to temperature with a transfer function of 20 mV/oC. The sensor has an output resistance of 5 kΩ. If the temperature is 50oC, find the amplifier output.

Impedance Matching

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Dynamic Characteristics (self study) • Zero order transducers (e.g. Potentiometer)

Dynamic Characteristics (self study) • First order transducers (e.g. Thermocouple)

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Dynamic Characteristics (self study) • Second order transducers (e.g. Accelerometer)

Dynamic Characteristics (self study) • Second order transducers

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Dynamic Characteristics (self study) • Second order transducers

1. Delay time (td) 2. Rise time (tr) 3. Peak time (tp) 4. Maximum overshoot (MP) 5. Settling time (ts)

Dynamic Characteristics (self study) • Second order transducers

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Generalised Model of a System Element • If hysteresis and resolution effects are not present in an element but environmental and non-linear effects are, then the steady-state output O of the element is in general given by:

Where: I: System input K:static sensitivity a: percept

KM: the change in sensitivity for unit change in IM

KI: the change in zero bias for unit change in II N(I): Non-linear error

Km and KI: environmental coupling constants (sensitivity and zero drift)

IM and II: Deviation from standard conditions (sensitivity and zero drift)

Generalised Model of a System Element

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Generalised Model of a System Element

Statistical variations in the output of a single element with time (Repeatability) • (Repeatability is the ability of an element to give the same output for the same input, when repeatedly applied to it. Lack of repeatability is due to random effects in the element and its environment. • The most common cause of lack of repeatability in the output O is random fluctuations with time in

the environmental inputs IM, II. If the coupling constants KM, KI are non-zero, then there will be corresponding time variations in O.

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Statistical variations in the output of a single element with time (Repeatability)

• By making reasonable assumptions for the probability

density functions of the inputs I, IM and II the probability density function of the element output O can be found.

• The most likely probability density function for I, IM and II is the normal or Gaussian distribution function

Statistical variations in the output of a single element with time (Repeatability)

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Statistical variations in the output of a single element with time (Repeatability)

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Statistical variations in the output of a single element with time (Repeatability)

Summary • In the general case of a batch of several ‘identical’ elements, where each element is subject to random variations in environmental conditions with time,

both inputs I, IM and II and parameters K, a, etc., are subject to statistical variations. • If we assume that each statistical variation can be represented by a normal probability density function, then the probability density function of the element output O is also normal, i.e.:

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Summary

Calibration (self reading) • If tolerances of a sensor and interface circuit are broader than the required overall accuracy, a calibration of the sensor or, preferably, a combination of a sensor and its interface circuit is required for minimizing errors. • A calibration requires application of several precisely known stimuli and reading the corresponding sensor responses. These are called the calibration points whose input–output values are the point coordinates. (Handbook of modern sensors)

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The Accuracy of Measurement Systems

Measurement error of a system of ideal elements • Consider the system shown in the figure consisting of n elements in series. Suppose each element is ideal, i.e. perfectly linear and not subject to environmental inputs. If we also assume the intercept or bias is zero, i.e. a = 0, then:

The Accuracy of Measurement Systems

Measurement error of a system of ideal elements • If the measurement system is complete, then E = O − I, giving:

• Thus, if we have E = 0 and the system is perfectly accurate. • Example:

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The Accuracy of Measurement Systems Measurement error of a system of ideal elements • The temperature measurement system shown in the Figure appears to satisfy the above condition. • However, The system is not accurate because none of the three elements present is ideal. The thermocouple is non-linear so that as the input temperature changes the sensitivity is no longer 40 μV /°C. Also changes in reference junction temperature cause the thermocouple e.m.f. to change. The output voltage of the amplifier is also affected by changes in ambient temperature. • In general, the error of any measurement system depends on the non-ideal characteristics – e.g. non-linearity, environmental and statistical effects – of every element in the system.

The Accuracy of Measurement Systems

The error probability density function of a system of non-ideal elements • the probability density function of the output p(O), the mean value of the distribution, and the standard

deviation σ0 were previously determined by equations for single element. • These equations apply to each element in a measurement system of n elements and can be used to calculate the system error probability density function p(E)

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The Accuracy of Measurement Systems The error probability density function of a system of non-ideal elements

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The Accuracy of Measurement Systems

The error probability density function of a system of non-ideal elements

Error Reduction Techniques

• The error of a measurement system depends on the non-ideal characteristics of every element in the system. • Using the calibration techniques, we can identify which elements in the system have the most dominant non-ideal behaviour. • We can then develop compensation strategies for these elements which should produce significant reductions in the overall system error. This section outlines compensation methods for non-linear and environmental effects.

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Error Reduction Techniques

• Compensating non-linear element • Isolation • Zero environmental sensitivity • Opposing environmental inputs • Differential system • High-gain negative feedback • Computer estimation of measured value

Error Reduction Techniques Compensating non-linear element

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Error Reduction Techniques Isolation • The most obvious method of reducing the effects of environmental inputs is that of isolation, i.e. to isolate the transducer from environmental changes

so that effectively IM = II = 0. • Examples of this are the placing of the reference junction of a thermocouple in a temperature- controlled enclosure.

Error Reduction Techniques Zero environmental sensitivity • Another obvious method is that of zero environmental sensitivity, where the element is completely insensitive to environmental inputs, i.e. KM = KI = 0. • An example of this is the use of a metal alloy with zero temperature coefficients of expansion and resistance as a strain gauge element. Such an ideal material is difficult to find, and in practice the resistance of a metal strain gauge is affected slightly by changes in ambient temperature.