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.
- dimension electrical voltage 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.
- the resistance of the resistor is found on the basis of 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 non-identical quantities, to find the relationship between them. In this case, the system of equations is solved.
- determination of the dependence of resistance on temperature. At the same time, non-similar quantities are measured, and the dependence is determined based on the measurement results.
Cumulative dimension
Cumulative dimension- simultaneous measurement of several quantities of the same name, in which the desired values of quantities are found by solving a system of equations consisting of the resulting direct measurements various combinations these quantities.
- measurement of the resistance of resistors connected by a triangle. In this case, the resistance value between the vertices is measured. Based on the results, the resistances of the resistors are determined.
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Calculation of errors for direct and indirect measurements
Measurement is understood as a comparison of the measured value with another value, taken as a unit of measurement. Measurements are carried out empirically using special technical means.
Direct measurements are called measurements, the result of which is 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 the quantities whose values are obtained in the process of direct measurements (for example, determining the 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 performed, is necessarily accompanied by an error (error) - a deviation of the measurement result from the true value of the measured quantity.
Systematic errors are errors, the magnitude of which 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 deviate from zero division in the absence of current; the balance beam may have unequal arms, etc.);
As a result of insufficient development of the theory of the measurement method, i.e., the measurement method contains a source of errors (for example, an error occurs when heat loss in environment or when weighing on an analytical balance is performed without taking into account the buoyancy of the air);
As a result of the fact that the change in the conditions of the experiment is not taken into account (for example, during the 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 if the features of the instruments are studied, the theory of experiment is developed more fully, and on the basis of this, corrections are made 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 senses, and in many other circumstances that accompany measurements, and which cannot be taken into account in advance (random errors occur, for example, if the equality of illumination fields of the photometer is set by eye; if the moment of maximum deviation mathematical pendulum determined by eye; when finding the moment of sound resonance by ear; when weighing on an analytical balance, if vibrations of the floor and walls are transmitted to the balance, 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 the same values were obtained when repeating the measurements, then this indicates not the absence of random errors, but 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.
Knowingly incorrect results - misses occur due to violation of the basic conditions of measurement, as a result of inattention or negligence of the experimenter. For example, in poor lighting, instead of “3”, write “8”; due to the fact that the experimenter is distracted, he can go astray when counting the number of swings of the pendulum; due to carelessness or inattention, he may confuse the masses of the masses when determining the spring constant, etc. external sign a miss is a sharp difference in magnitude from the results of other measurements. If a miss is detected, the measurement result should be discarded immediately, and the measurement itself should be repeated. The identification of blunders is also aided by a comparison of the measurement results obtained by different experimenters.
To measure a physical quantity means to find the confidence interval in which it lies true value 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 falls within the confidence interval. The value is expressed either in fractions of a unit or as a percentage. For most measurements, they are limited to a confidence probability of 0.9 or 0, 95. Sometimes, when it is required extremely high degree reliability, set a confidence level of 0.999. Along with the confidence level, a significance level is often used, which specifies the probability that the true value does not fall within the confidence interval. The measurement result is presented in the form
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 range.
To find and , perform a series of single measurements. 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 .
Mean value of measured value
The measuring device also contributes to the measurement error. 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 the magnitude nor the sign of it for this particular instrument is known. The instrumental error is evaluated in the process of testing a large series of the same type of instruments.
The normalized 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 the device is equal to the relative error of the device, expressed as a percentage, in relation to the full range of the scale. Passport error of the device
Direct measurements called such measurements that are obtained directly with the help of measuring device. Direct measurements include measuring length with a ruler, calipers, measuring voltage with a voltmeter, measuring temperature with a thermometer, etc. Various factors can influence the results of direct measurements. Therefore, the measurement error has a different form, i.e. there is an instrument error, systematic and random errors, rounding errors when reading off the instrument scale, misses. In this regard, it is important to identify in each particular experiment which of the measurement errors is the largest, and if it turns out that one of them is an order of magnitude higher than all the others, then the last errors can be neglected.
If all the considered errors are of the same order of magnitude, then it is necessary to evaluate the combined effect of several different errors. In the general case, the total error is calculated by 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 by means of formulas. Such measurements are called indirect. In the language of mathematics, this means that the desired physical quantity f associated with 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 value is V- ball, which will be determined by direct measurement of the radius of the ball 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 value, 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)
The theory of errors shows that the error of a function is estimated by the sum of the errors of all arguments. The error of the function will be the smaller, the smaller the errors of its arguments.
4. Construction of graphs for experimental measurements.
An essential point of the experimental study is the construction of graphs. When plotting graphs, first of all, you need to choose a coordinate system. The most common is a rectangular coordinate system with a coordinate grid formed by parallel lines equidistant from each other (for example, graph paper). On the coordinate axes, divisions are applied at certain intervals on a certain scale for the function and argument.
In laboratory work, when studying physical phenomena, one has 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 from temperature. Many more examples could be cited.
variable At is called a function of another variable X(argument) if each value At will correspond to a well-defined value of the quantity X, then we can write the dependence of the function in the form Y \u003d Y (X).
It follows from the definition of the function that to define it, it is necessary to specify two sets of numbers (argument values X and features At), 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 way of setting the function - the dependence of the value of direct current I on the magnitude of the voltage U, i.e. I= f(U) .
table 2
2. The analytical way 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 way of setting the function.
Function Graph I= f(U) in the Cartesian coordinate system is called the locus of points, built on numerical values coordinate point arguments and functions.
On fig. 1 built dependency graph I= f(U) , given by the table.
The points found in the experiment and plotted on the graph are clearly marked in the form of circles and crosses. On the graph, for each constructed point, it is necessary to indicate the errors in the form of "hammers" (see Fig. 1). The sizes of these "hammers" should be equal to twice the value of 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 would be no 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 the argument or function is separated from the origin by a value comparable to the value of the interval itself, then it is advisable to move the origin to a point close to the beginning of the interval under study, both along the abscissa and along the ordinate.
Drawing a curve (i.e., connecting experimental points) through points is usually carried out in accordance with the ideas of the least squares method. In probability theory, it is shown that the best approximation to the experimental points will be such a curve (or straight line) for which the sum of the least squares of deviations along the vertical from the point to the curve will be minimal.
The points put 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 of not exceeded errors and 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 go beyond the range of permissible values (see Fig. 2, points BUT and AT), then the curve is drawn along the remaining points, and the dropped points BUT and AT as misses are not taken into account. Then repeated measurements are taken in this area (points BUT and AT) and the reason for such a deviation is established (either this is a mistake or a legitimate violation of the found dependence).
If the investigated, experimentally constructed function detects "special" points (for example, points of extremum, inflection, break, etc.). This increases the number of experiments at small values of the step (argument) in the region of singular points.
The classification of types of measurements can be carried out according to various classification criteria, which include the following:
Way of finding numerical value physical quantity,
Number of observations,
The nature of the dependence of the measured value on time,
The number of measured instantaneous values in a given time interval,
Conditions that determine the accuracy of the results,
A way of expressing measurement results.
By method of finding the numerical value of a physical quantity measurements are divided into the following types: direct, indirect,aggregate and joint.
Direct measurement called a measurement in which the value of the measured quantity is found directly from the experimental data. Direct measurements are carried out using means designed to measure these quantities. The numerical value of the measured value is read directly from the indication of the measuring device. Examples of direct measurements: current measurement with an ammeter; voltage - voltmeter; masses - on lever scales, etc.
The relationship between the measured value X and the measurement result Y in direct measurement is characterized by the equation:
those. the value of the measured quantity is taken equal to the result obtained.
Unfortunately, direct measurement is not always possible. Sometimes there is no appropriate measuring device at hand or it is unsatisfactory in accuracy, or even has not yet been created at all. In this case, one has to resort to indirect measurement.
By indirect measurements called such 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 quantity itself that is measured, but other quantities that are functionally related to it. The value of the quantity measured indirectly X find by calculation by the formula
X=F(Y 1 , Y 2 , … , Y n),
where Y 1 , Y 2 , … Y n are the 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 values \u200b\u200bof the voltage drop 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.
Cumulative measurements - these are simultaneous measurements of several quantities of the same name, in which the desired values \u200b\u200bof 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 \u200b\u200bof resistors connected by a triangle (Fig. 3.1), the resistances are measured at each pair of vertices of the triangle and a system of equations is obtained:
 |
From the solution of this system of equations, the resistance values are obtained
, 
, 
,
Joint measurements- these are simultaneous measurements of two or more quantities that are not of the same name 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 are the values of the required quantities.
For example, the inductance of the coil
L = L 0 ×(1 + w 2 × C × L 0),
where L0– inductance at frequency w =2×p×f tending to zero; With- interturn capacitance. Values L0 and With cannot be found by direct or indirect measurements. Therefore, in the simplest case, measure L1 at w 1, and then L2 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, find the desired values of inductance L0 and containers With
; 
.
Cumulative and joint measurements are a generalization of indirect measurements to the case of several quantities.
To improve the accuracy of cumulative 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 sought quantities. The resulting inconsistent system of equations is solved by the least squares method.
By number of measurements subdivided:
On the ordinary measurements – measurements performed with a single observation;
- statistical measurements – measurements with multiple observations.
Observation in measurement - an experimental operation performed in the course of measurements, 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 the quantity obtained in a separate observation.
By the nature of the dependence of the measured value on time measurements are separated:
On the static , at which the measured value remains constant in time during the measurement process;
- dynamic , at which the measured value changes during the measurement process and is not constant in time.
With dynamic measurements, this change must be taken into account in order to obtain a measurement result. And to assess the accuracy of the results of dynamic measurements, it is necessary to know the dynamic properties of measuring instruments.
According to 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 the number of measured instantaneous values is finite in a given time interval.
Continuous (analogue) measurements are measurements in which the number of measured instantaneous values in a given time interval is infinite.
According to the conditions that determine the accuracy of the results, measurements are:
- highest possible accuracy achievable with the current state of the art;
- control and calibration, the error of which should not exceed some set 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) the use of values of physical constants.
Relative measurements - measuring the ratio of a quantity to the same-named value, which plays the role of a unit, or measuring the value in relation to the same-named value, taken as the initial one.
Measurement methods and their classification
All measurements can be made by various methods. There are two main measurement methods: direct evaluation method and comparison methods with measure.
Direct evaluation method characterized by the fact that the value of the measured quantity is determined directly by the reading device of the measuring instrument, pre-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 balance, force electric current pointer ammeter, phase difference digital phase meter, etc.
The functional diagram of the measurement by the method of direct evaluation is shown in fig. 3.2.
The measure in instruments of direct evaluation is the division of the scale of the reading device. They are set not arbitrarily, but on the basis of the calibration of the device. Thus, the divisions of the scale of the reading device are, as it were, a substitute (²imprint²) for 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 evaluation devices actually implement the principle of comparison with physical quantities. But this comparison is different and is carried out indirectly, with the help of an intermediate means - divisions of the scale of the reading device.
Measure Comparison Methods – methods of measurement in which the quantity being measured is compared with the quantity reproducible by the measure. These methods are more accurate than the direct estimation method, but are slightly more complicated. The group of comparison methods with measure includes the following methods: opposition method, null method, differential method, the match method, and the replacement method.
defining feature 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 "imprints".
The comparison can be simultaneous and different. With simultaneous comparison, the measure and the measured value act on the measuring device at the same time, and when multi-temporal– the impact of the measured quantity and the measure on the measuring device is separated in time. Moreover, the comparison can be immediate and indirect.
In direct comparison, the measured value and the measure directly affect the comparison device, and in indirect comparison, through other quantities that are unambiguously related to the known and measured values.
Simultaneous comparison is usually carried out by methods opposition, zero, differential and coincidences, and multi-temporal - substitution method.
LECTURE 4
MEASUREMENT METHODS
Indirect measurements differ from direct ones in that the desired value of the quantity is determined on the basis of the results of direct measurements of other physical. values functionally related to the desired value. In other words, the desired value of the PV is established based on the results of direct measurements of such quantities that are associated with the desired specific dependence. Indirect measurement equation: y \u003d f (x 1, x 2, ..., x p), where x i - i is the th result of direct measurement. Examples: In modern microprocessor-based measuring devices, very often the calculations of the required measured value are carried out "inside" the device. In this case, the measurement result is determined by the method typical for direct measurements, and there is no need and possibility of separately taking into account the methodological error of the calculation. It is included in the error of the measuring device. Measurements carried out by measuring instruments of this kind are direct. Indirect measurements include only those measurements in which the calculation is carried out manually or automatically, but after receiving the results of direct measurements. In this case, the calculation error can be taken into account separately. An example of such a case is measuring systems for which the metrological characteristics of their components are normalized separately. The total measurement error is calculated according to the normalized metrological characteristics of all system components. Cumulative measurements are associated with the solution of a system of equations compiled from the results of simultaneous measurements of several homogeneous quantities. The solution of the system of equations makes it possible to calculate the desired value.
In cumulative measurements, the values of a set of similar quantities Q 1 ... ... Q k ., as a rule, are determined by measuring the sums or differences of these quantities in various combinations:
where the coefficients c ij take the values ±1 or 0.
Thus, we are talking about simultaneous measurements of several quantities of the same name, in which the desired values of the quantities are determined by solving a system of equations obtained by measuring various combinations of these quantities.
Joint measurements- these are simultaneous (direct or indirect) measurements of two or more inhomogeneous (not of the same name) physical. quantities to determine the functional relationship between them. In fact, aggregate measurements are no different from joint measurements, except that in the first case, the measurements refer to the quantities of the same name, and in the second, to non-similar ones. Indirect, aggregate and joint measurements are united by one fundamentally important common property: their results are determined by the calculation of known functional dependencies between the measured quantities and the quantities subjected to direct measurements.
Thus, we emphasize once again that the difference between indirect, cumulative and joint measurements is only in the form of a function dependence used in the calculations. With indirect measurements, it is expressed by one equation in explicit form, with joint and cumulative measurements, it is expressed by a system of implicit equations.
