Calculation of errors in direct and indirect measurements
Measurement refers to the comparison of a measured quantity with another quantity taken as a unit of measurement. Measurements are carried out experimentally using special technical means.
Direct measurements are measurements whose results are obtained directly from experimental data (for example, measuring length with a ruler, time with a stopwatch, temperature with a thermometer). Indirect measurements are measurements in which the desired value of a quantity is found on the basis of a known relationship between this quantity and quantities whose values are obtained in the process of direct measurements (for example, determining speed along the distance traveled and time https://pandia.ru/text/78/ 464/images/image002_23.png" width="65" height="21 src=">).
Any measurement, no matter how carefully it is carried out, is necessarily accompanied by an error (error) - a deviation of the measurement result from the true value of the measured value.
Systematic errors are errors whose magnitude is the same in all measurements carried out by the same method using the same measuring instruments, under the same conditions. Systematic errors occur:
As a result of the imperfection of the instruments used in measurements (for example, the ammeter needle may be deviated from the zero division in the absence of current; the balance beam may have unequal arms, etc.);
As a result, the theory of the measurement method is not fully developed, i.e. the measurement method contains a source of errors (for example, an error occurs when heat loss to the environment is not taken into account in calorimetric work or when weighing on an analytical balance is carried out without taking into account the buoyant force of air) ;
As a result of the fact that changes in experimental conditions are not taken into account (for example, during long-term passage of current through the circuit, as a result of the thermal effect of the current, the electrical parameters of the circuit change).
Systematic errors can be eliminated by studying the features of instruments, more fully developing the theory of experience, and based on this, making corrections to the measurement results.
Random errors are errors whose magnitude is different even for measurements made in the same way. Their reasons lie both in the imperfection of our sense organs and in many other circumstances accompanying measurements, and which cannot be taken into account in advance (random errors arise, for example, if the equality of illumination of the photometer fields is established by eye; if the moment of maximum deflection of a mathematical pendulum is determined by eye ; when finding the moment of sound resonance by ear; when weighing on analytical balances, if vibrations of the floor and walls are transmitted to the scales, etc.).
Random errors cannot be avoided. Their occurrence is manifested in the fact that when repeating measurements of the same quantity with the same care, numerical results are obtained that differ from each other. Therefore, if, when repeating measurements, the same values were obtained, this does not indicate the absence of random errors, but rather the insufficient sensitivity of the measurement method.
Random errors change the result both in one direction and in the other direction from the true value, therefore, in order to reduce the influence of random errors on the measurement result, measurements are usually repeated many times and the arithmetic mean of all measurement results is taken.
Deliberately incorrect results - errors arise due to violation of the basic measurement conditions, as a result of the inattention or negligence of the experimenter. For example, in poor lighting, “8” is written instead of “3”; due to the fact that the experimenter is distracted, he may get confused when counting the number of oscillations of the pendulum; due to negligence or inattention, he may confuse the masses of the loads when determining the stiffness of the spring, etc. An external sign of a mistake is a sharp difference in the value of the result from the results of other measurements. If a mistake is detected, the measurement result should be immediately discarded and the measurement itself should be repeated. The identification of errors is also facilitated by comparison of measurement results obtained by different experimenters.
To measure a physical quantity means to find a confidence interval in which its true value lies https://pandia.ru/text/78/464/images/image005_14.png" width="16 height=21" height="21">. .png" width="21" height="17 src=">.png" width="31" height="21 src="> cases, the true value of the measured value will fall into the confidence interval. The value is expressed either in fractions of a unit, or in percent. For most measurements, the confidence level is limited to 0.9 or 0.95. Sometimes, when an extremely high degree of reliability is required, a confidence level of 0.999 is set. Along with the confidence level, a significance level is often used, which specifies the probability that the true value does not fall into the range. confidence interval. The measurement result is presented as
where https://pandia.ru/text/78/464/images/image012_8.png" width="23" height="19"> is the absolute error. Thus, the boundaries of the interval, https://pandia.ru/ text/78/464/images/image005_14.png" width="16" height="21"> lies within this interval.
In order to find and , a series of single measurements is performed. Let's consider a specific example..png" width="71" height="23 src=">; ; https://pandia.ru/text/78/464/images/image019_5.png" width="72" height=" 23">.png" width="72" height="24">. Values can be repeated, like values and https://pandia.ru/text/78/464/images/image024_4.png" width="48 height=15" height="15">.png" width="52" height="21">. Accordingly, the significance level is .
Average value of the measured quantity
The measuring instrument also contributes to measurement uncertainty. This error is due to the design of the device (friction in the axis of the pointer device, rounding produced by a digital or discrete pointer device, etc.). By its nature, this is a systematic error, but neither its magnitude nor sign is known for this particular device. Instrument error is assessed during testing of a large series of similar devices.
The standardized range of accuracy classes of measuring instruments includes the following values: 0.05; 0.1; 0.2; 0.5; 1.0; 1.5; 2.5; 4.0. The accuracy class of an instrument is equal to the relative error of the instrument expressed as a percentage relative to the full scale range. Specification error of the device
Indirect measurement
Direct measurement
Direct measurement- this is a measurement in which the desired value of a physical quantity is found directly from experimental data as a result of comparing the measured quantity with standards.
- measuring length with a ruler.
- measuring electrical voltage with a voltmeter.
Indirect measurement
Indirect measurement- a measurement in which the desired value of a quantity is found on the basis of a known relationship between this quantity and the quantities subjected to direct measurements.
- We find the resistance of the resistor based on Ohm's law by substituting the values of current and voltage obtained as a result of direct measurements.
Joint measurement
Joint measurement- simultaneous measurement of several different quantities to find the relationship between them. In this case, a system of equations is solved.
- determination of the dependence of resistance on temperature. In this case, different quantities are measured, and the dependence is determined based on the measurement results.
Aggregate Measurement
Aggregate Measurement- simultaneous measurement of several quantities of the same name, in which the desired values of the quantities are found by solving a system of equations consisting of the resulting direct measurements of various combinations of these quantities.
- measuring the resistance of resistors connected in a triangle. In this case, the resistance value between the vertices is measured. Based on the results, the resistor resistances are determined.
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According to the method of obtaining the values of a physical quantity measurements can be direct, indirect, cumulative and joint, each of which is carried out using absolute and relative methods (see clause 3.2.).
Rice. 3. Classification of types of measurements
Direct measurement– a measurement in which the desired value of a quantity is found directly from experimental data. Examples of direct measurements are determining length using linear measures or determining temperature with a thermometer. Direct measurements form the basis of more complex indirect measurements.
Indirect measurement – measurement in which the desired value of a quantity is found on the basis of a known relationship between this quantity and quantities obtained by direct measurements, for example, trigonometric methods of measuring angles, in which the acute angle of a right triangle is determined from the measured lengths of the legs and hypotenuse, or measuring the average diameter of a thread using the three-wire method or, the power of an electrical circuit based on the voltage measured by a voltmeter and current measured by an ammeter, using a known dependence. In some cases, indirect measurements provide more accurate results than direct measurements. For example, the errors in direct measurements of angles using goniometers are an order of magnitude higher than the errors in indirect measurements of angles using sine rulers.
Joint are measurements made simultaneously of two or more opposite quantities. The purpose of these measurements is to find a functional relationship between quantities.
Example 1. Construction of a calibration characteristic y = f(x) measuring transducer, when sets of values are simultaneously measured:
X 1, X 2, X 3, …, X i, …, X n
Y 1, Y 2, Y 3, …, Y i, …, Y n
Example 2. Determination of the temperature coefficient of resistance by simultaneous resistance measurements R and temperature t and then defining the dependency a(t) = DR/Dt:
R 1 , R 2 , …, R i , …, R n
t 1 , t 2 , …, t i , …, t n
Aggregate Measurements are carried out by simultaneous measurement of several quantities of the same name, at which the desired value is found by solving a system of equations obtained as a result of direct measurements of various combinations of these quantities.
Example: the mass value of the individual weights of the set is determined from the known value of the mass of one of the weights and from the results of measurements (comparisons) of the masses of various combinations of weights.
There are weights with masses m 1, m 2, m 3.
The mass of the first weight is determined as follows:
The mass of the second weight will be determined as the difference between the masses of the first and second weights M 1.2 and the measured mass of the first weight:
The mass of the third weight will be determined as the difference in the mass of the first, second and third weights ( M 1,2,3) and measured masses of the first and second weights ():
Often this is the way to improve the accuracy of measurement results.
Cumulative measurements differ from joint ones only in that with cumulative measurements several quantities of the same name are measured simultaneously, and with joint measurements they measure different quantities.
Cumulative and joint measurements are often used when measuring various parameters and characteristics in the field of electrical engineering.
By the nature of the change in the measured quantity There are static, dynamic and statistical measurements.
Static– measurements of PVs that do not change over time, for example, measuring the length of a part at normal temperature.
Dynamic– measurements of time-varying PV, for example measuring distance to ground level from a descending aircraft, or voltage in an alternating current network.
Statistical measurements are associated with determining the characteristics of random processes, sound signals, noise levels, etc.
By accuracy There are measurements with the highest possible accuracy, control and verification and technical.
Measurements with the highest possible accuracy– these are reference measurements related to the accuracy of reproducing units of physical quantities, measurements of physical constants. These measurements are determined by the current state of the art.
Control and verification– measurements, the error of which should not exceed a certain specified value. These include measurements performed by laboratories of state supervision over the implementation and compliance with standards and the state of measuring equipment, measurements by factory measurement laboratories and others, carried out using means and techniques that guarantee an error not exceeding a predetermined value.
Technical measurements– measurements in which the error of the result is determined by the characteristics of measuring instruments (MI). This is the most widespread type of measurement, carried out using working measuring instruments, the error of which is known in advance and is considered sufficient to perform this practical task.
Measurements by way of expressing measurement results can also be absolute and relative.
Absolute measurement– a measurement based on direct measurements of one or more basic quantities, as well as on the use of values of physical constants. In linear and angular absolute measurements, as a rule, one physical quantity is found, for example, the diameter of a shaft using a caliper. In some cases, the values of the measured quantity are determined by direct reading on the scale of the device, calibrated in units of measurement.
Relative dimension– measurement of the ratio of a quantity to a quantity of the same name, which plays the role of a unit. At relative method measurements, the value of the deviation of the measured value relative to the size of the installation standard or sample is assessed. An example is measurement on an optimometer or minimeter.
By number of measurements a distinction is made between single and multiple measurements.
Single measurements– this is one measurement of one quantity, i.e. the number of measurements is equal to the number of measured quantities. The practical application of this type of measurement is always associated with large errors, so at least three single measurements should be carried out and the final result should be found as the arithmetic mean value.
Multiple measurements characterized by an excess of the number of measurements of the number of measured quantities. Usually the minimum number of measurements in this case is more than three. The advantage of multiple measurements is a significant reduction in the influence of random factors on the measurement error.
The types of measurements given include various methods, i.e. methods for solving the measurement problem with theoretical justification according to the accepted methodology.
Direct measurements These are measurements that are obtained directly using a measuring device. Direct measurements include measuring length with a ruler, calipers, measuring voltage with a voltmeter, measuring temperature with a thermometer, etc. The results of direct measurements can be influenced by various factors. Therefore, the measurement error has a different form, i.e. There are instrument errors, systematic and random errors, rounding errors when taking readings from the instrument scale, and misses. In this regard, it is important to identify in each specific experiment which of the measurement errors is the largest, and if it turns out that one of them is an order of magnitude greater than all the others, then the latter errors can be neglected.
If all the errors taken into account are the same order of magnitude, then it is necessary to evaluate the combined effect of several different errors. In general, the total error is calculated using the formula:
Where – random error, – instrument error, – rounding error.
In most experimental studies, a physical quantity is measured not directly, but through other quantities, which in turn are determined by direct measurements. In these cases, the measured physical quantity is determined through directly measured quantities using formulas. Such measurements are called indirect. In the language of mathematics, this means that the desired physical quantity f related to other quantities X 1, X 2, X 3, … ,. X n functional dependence, i.e.
F= f(x 1 , x 2 ,….,X n )
An example of such dependencies is the volume of a sphere
.
In this case, the indirectly measured quantity is V- ball, which is determined by direct measurement of the ball radius R. This measured value V is a function of one variable.
Another example would be the density of a solid
.
(8)
Here – is an indirectly measured quantity, which is determined by direct measurement of body weight m and indirect value V. This measured value is a function of two variables, i.e.
= (m, V)
Error theory shows that the error of a function is estimated by the sum of the errors of all arguments. The smaller the errors of its arguments, the smaller the error of a function.
4. Plotting graphs based on experimental measurements.
An essential point of experimental research is the construction of graphs. When constructing graphs, first of all you need to select a coordinate system. The most common is a rectangular coordinate system with a coordinate grid formed by equally spaced parallel lines (for example, graph paper). On the coordinate axes, divisions are marked at certain intervals on a certain scale for the function and argument.
In laboratory work, when studying physical phenomena, it is necessary to take into account changes in some quantities depending on changes in others. For example: when considering the movement of a body, a functional dependence of the distance traveled on time is established; when studying the electrical resistance of a conductor as a function of temperature. Many more examples can be given.
Variable value U called a function of another variable X(argument) if each has a value U will correspond to a very specific value of the quantity X, then we can write the dependence of the function in the form Y = Y(X).
From the definition of the function it follows that to specify it it is necessary to specify two sets of numbers (argument values X and functions U), as well as the law of interdependence and correspondence between them ( X and Y). Experimentally, the function can be specified in four ways:
Table; 2. Analytically, in the form of a formula; 3. Graphically; 4. Verbally.
For example: 1. Tabular method of specifying the function - dependence of the magnitude of direct current I on the voltage value U, i.e. I= f(U) .
table 2
2.The analytical method of specifying a function is established by a formula, with the help of which the corresponding values of the function can be determined from the given (known) values of the argument. For example, the functional dependence shown in Table 2 can be written as:
(9)
3. Graphical method of specifying a function.
Function graph I= f(U) in the Cartesian coordinate system is the geometric locus of points constructed from the numerical values of the coordinate point of the argument and function.
In Fig. 1 plotted dependence I= f(U) , specified by the table.
Points found experimentally and plotted on a graph are clearly marked as circles and crosses. On the graph, for each plotted point, it is necessary to indicate errors in the form of “hammers” (see Fig. 1). The size of these “hammers” should be equal to twice the absolute errors of the function and argument.
The scales of the graphs must be chosen so that the smallest distance measured from the graph is not less than the largest absolute measurement error. However, this choice of scale is not always convenient. In some cases, it is more convenient to take a slightly larger or smaller scale along one of the axes.
If the studied interval of values of an argument or function is distant from the origin of coordinates by an amount comparable to the value of the interval itself, then it is advisable to move the origin of coordinates to a point close to the beginning of the studied interval, both along the abscissa and ordinate axis.
Fitting a curve (i.e., connecting experimental points) through points is usually done in accordance with the ideas of the method of least squares. In probability theory, it is shown that the best approximation to experimental points will be a curve (or straight line) for which the sum of the least squares of vertical deviations from the point to the curve will be minimal.
The points marked on the coordinate paper are connected by a smooth curve, and the curve should pass as close as possible to all experimental points. The curve should be drawn so that it lies as close as possible to the points where the errors are not exceeded and so that there are approximately equal numbers of them on both sides of the curve (see Fig. 2).
If, when constructing a curve, one or more points fall outside the range of permissible values (see Fig. 2, points A And IN), then the curve is drawn along the remaining points, and the dropped points A And IN how misses are not taken into account. Then repeated measurements are taken in this area (points A And IN) and the reason for such a deviation is established (either it is a mistake or a legal violation of the found dependence).
If the studied, experimentally constructed function detects “special” points (for example, points of extremum, inflection, discontinuity, etc.). Then the number of experiments increases at small values of the step (argument) in the region of singular points.
Classification of types of measurements can be carried out according to various classification criteria, which include the following:
A method for finding the numerical value of a physical quantity,
Number of observations
The nature of the dependence of the measured quantity on time,
The number of measured instantaneous values in a given time interval,
Conditions that determine the accuracy of the results
Method of expressing measurement results.
By method of finding the numerical value of a physical quantity measurements are divided into the following types: direct, indirect,cumulative and joint.
Direct measurement called a measurement in which the value of the measured quantity is found directly from experimental data. Direct measurements are performed using tools designed to measure these quantities. The numerical value of the measured quantity is calculated directly from the reading of the measuring device. Examples of direct measurements: current measurement with an ammeter; voltage - with a voltmeter; mass - on lever scales, etc.
The relationship between the measured value X and the measurement result Y during direct measurement is characterized by the equation:
those. the value of the measured quantity is assumed to be equal to the result obtained.
Unfortunately, direct measurement is not always possible. Sometimes the appropriate measuring instrument is not at hand, or it is unsatisfactory in accuracy, or has not even been created yet. In this case, you have to resort to indirect measurement.
Indirect measurements These are measurements in which the value of the desired quantity is found on the basis of a known relationship between this quantity and the quantities subjected to direct measurements.
In indirect measurements, it is not the actual quantity being determined that is measured, but other quantities that are functionally related to it. The value of the quantity measured indirectly X found by calculation using the formula
X = F(Y 1 , Y 2 , … , Y n),
Where Y 1 , Y 2 , … Y n– values of quantities obtained by direct measurements.
An example of an indirect measurement is the determination of electrical resistance using an ammeter and a voltmeter. Here, by direct measurements, the voltage drop values are found U on resistance R and current I through it, and the desired resistance R is found by the formula
R = U/I.
The operation of calculating the measured value can be performed by both a person and a computing device placed in the device.
Direct and indirect measurements are currently widely used in practice and are the most common types of measurements.
Aggregate Measurements – these are measurements of several quantities of the same name made simultaneously, in which the desired values of the quantities are found by solving a system of equations obtained by direct measurements of various combinations of these quantities.
For example, to determine the resistance values of resistors connected by a triangle (Fig. 3.1), the resistances at each pair of vertices of the triangle are measured and a system of equations is obtained:
 |
From the solution of this system of equations the resistance values are obtained
, 
, 
,
Joint measurements– these are measurements of two or more quantities of the same name that are made simultaneously X 1, X 2,…,X n, whose values are found by solving the system of equations
F i(X 1, X 2, …, X n; Y i1 , Y i2 , … ,Y im) = 0,
Where i = 1, 2, …, m > n; Y i1 , Y i2 , … ,Y im– results of direct or indirect measurements; X 1, X 2, …, X n– values of the required quantities.
For example, the inductance of the coil
L = L 0 ×(1 + w 2 × C × L 0),
Where L 0– inductance at frequency w =2×p×f tending to zero; WITH– interturn capacitance. Values L 0 And WITH cannot be found by direct or indirect measurements. Therefore, in the simplest case we measure L 1 at w 1, and then L 2 at w 2 and form a system of equations:
L 1 = L 0 ×(1 + w 1 2 × C× L 0);
L 2 = L 0 ×(1 + w 2 2 × C× L 0),
solving which, we find the required inductance values L 0 and containers WITH
; 
.
Cumulative and joint measurements are a generalization of indirect measurements to the case of several quantities.
To increase the accuracy of aggregate and joint measurements, the condition m ³ n is provided, i.e. the number of equations must be greater than or equal to the number of required quantities. The resulting inconsistent system of equations is solved by the least squares method.
By number of measurement observations are divided:
On ordinary measurements – measurements performed with a single observation;
- statistical measurements – measurements with multiple observations.
Observation during measurement - an experimental operation performed during the measurement process, as a result of which one value is obtained from a group of values of quantities that are subject to joint processing to obtain measurement results.
Observation result– the result of a quantity obtained from a separate observation.
By the nature of the dependence of the measured quantity on time dimensions are divided:
On static , in which the measured quantity remains constant over time during the measurement process;
- dynamic , in which the measured quantity changes during the measurement process and is not constant over time.
In dynamic measurements, this change must be taken into account to obtain the measurement result. And to assess the accuracy of the results of dynamic measurements, knowledge of the dynamic properties of measuring instruments is necessary.
Based on the number of measured instantaneous values in a given time interval, measurements are divided into discrete And continuous(analog).
Discrete measurements are measurements in which, over a given time interval, the number of measured instantaneous values is finite.
Continuous (analog) measurements – measurements in which, over a given time interval, the number of measured instantaneous values is infinite.
According to the conditions determining the accuracy of the results, measurements are:
- highest possible accuracy, achieved with the existing level of technology;
- control and verification, the error of which should not exceed a certain specified value;
- technical measurements, in which the error of the result is determined by the characteristics of the measuring instruments.
By way of expressing results distinguish between absolute and relative measurements.
Absolute measurements – measurements based on direct measurements of one or more basic quantities and (or) use of the values of physical constants.
Relative measurements – measuring the ratio of a quantity to a quantity of the same name, which plays the role of a unit, or measuring a quantity in relation to a quantity of the same name, taken as the initial one.
Measurement methods and their classification
All measurements can be made using various methods. There are two main measurement methods: direct assessment method And methods of comparison with a measure.
Direct assessment method characterized by the fact that the value of the measured quantity is determined directly from the reading device of the measuring device, previously calibrated in units of the measured quantity. This method is the simplest and therefore is widely used in measuring various quantities, for example: measuring body weight on a spring scale, electric current with a dial ammeter, phase difference with a digital phase meter, etc.
The functional diagram of measurement using the direct assessment method is shown in Fig. 3.2.
The measure in direct assessment instruments is the division of the scale of the reading device. They are not placed arbitrarily, but based on the calibration of the device. Thus, the divisions of the scale of the reading device are, as it were, a substitute (a “fingerprint”) of the value of a real physical quantity and therefore can be used directly to find the values of the quantities measured by the device. Consequently, all direct assessment devices actually implement the principle of comparison with physical quantities. But this comparison is multi-temporal and is carried out indirectly, using an intermediate means - divisions of the scale of the reading device.
Methods for comparison with a measure – measurement methods in which the measured value is compared with the value reproduced by the measure. These methods are more accurate than the direct assessment method, but a little more complicated. The group of methods for comparison with a measure includes the following methods: contrast method, zero method, differential method, coincidence method and substitution method.
Defining characteristic comparison methods is that in the process of measurement there is a comparison of two homogeneous quantities - a known (reproducible measure) and a measured one. When measuring by comparison methods, real physical measures are used, and not their “fingerprints”.
Comparison can be simultaneous and multi-simultaneous. With simultaneous comparison, the measure and the measured quantity act on the measuring device simultaneously, and with multi-temporal– the impact of the measured quantity and measure on the measuring device is separated in time. In addition, comparison can be direct And indirect.
In direct comparison, the measured quantity and measure directly affect the comparison device, and in indirect comparison, through other quantities that are uniquely related to the known and measured quantities.
Simultaneous comparison is usually carried out using methods oppositions, zero, differential And coincidences, and multi-temporal - by substitution method.
LECTURE 4
MEASUREMENT METHODS