Additive and multiplicative errors of measuring instruments. Additive and multiplicative errors

According to the dependence of the absolute error on the values of the measured value, errors are distinguished:

● additive ∆ a, independent of the measured value;

● multiplicative ∆ m, which are directly proportional to the measured value;

● non-linear ∆ n, having a non-linear dependence on the measured value.

These errors are mainly used to describe the metrological characteristics of the measuring instrument. The division of errors into additive, multiplicative and nonlinear is very important in solving the problem of normalization and mathematical description of SI errors.

Examples of additive errors - from a constant weight on the weighing pan, from inaccurate zeroing of the instrument arrow before measurement, from thermo-EMF in DC circuits. The reasons for the occurrence of multiplicative errors can be: a change in the amplifier gain, a change in the rigidity of the gauge sensor membrane or the device spring, a change in the reference voltage in a digital voltmeter.

These types of errors are sometimes also referred to as:

● additive ---- zero error;

● multiplicative ----- the error of the slope of the characteristic;

● nonlinear --------- nonlinearity error.

Due to the fact that the additive and multiplicative components of the error are characteristic of a measuring instrument, and in the range of measured values, then, based on the given true (actual) value of the linear size of a structural element (14.3 cm), let us assume that the used measuring instrument allows make measurements in the range from 0.1 cm to 25 cm, and it has a uniform average relative error of 12.7% for the entire scale, which is calculated using formula (2.5) in the 2nd section of this work. Based on the selected measurement range of the measuring instrument (0.1 cm - 25 cm), take from it, for example, 10 equidistant fixed (reference) values of the linear size of a structural element, including the specified true (actual) value equal to 14.3 meters. As a result, a number of measured reference values of linear dimensions L this i, used by the measuring instrument, will have the form: 2.5; 5; 7.5; ten; 12.5; 15; 17.5; twenty; 22.5; 25 (cm).

Using expression (2.5), it is possible to determine the values of the total absolute error for all terms of the series ( L this i), namely:

(3.1)

The calculated values of the total absolute error ∆ s i for all members of the series, taking into account the fulfillment of the rules for rounding off measurement results and measurement errors (given in Appendix 1), are presented in Table 3.1.

Table 3.1

The results of calculating the total, additive and multiplicative

absolute errors

Member number	L this i, m	, %	∆ with i, cm	Δ a, cm	Δ m, cm
	2,5	12,7	0,318	0,318
		12,7	0,635	0,318	0,318
	7,5	12,7	0,953	0,318	0,635
		12,7	1,270	0,318	0,952
	12,5	12,7	1,588	0,318	1,27
		12,7	1,905	0,318	1,587
	17,5	12,7	2,223	0,318	1,905
		12,7	2,540	0,318	2,222
	22,5	12,7	2,858	0,318	2,54
		12,7	3,175	0,318	2,857

Using the results of calculations of the total absolute error ∆ with i and a number of measured reference values of linear dimensions L this i, a graph is plotted (see Fig. 3.2) of the dependencies , at the same time the points on which it is constructed are approximated. On the axes of the graph, the initial and final values of the measuring range of the measuring instrument (Len = 2.5 cm and Lek = 25 cm) and the maximum value of the total error Δ s (Δ ck = 3.175 cm) are indicated.

Rice. 3.2. Total absolute error plot

The resulting graph (Fig. 3.2) highlights the additive component (Δ a) of the total absolute error (Δ c), which is equal to the total absolute error at the minimum (initial) value of the reference values of linear dimensions (at the beginning of the SI measurement range), i.e. Δ a = 0.318 cm.

A graph (Fig. 3.3) of the dependence of the absolute additive error Δ a = f(L THIS. i), which is a straight line parallel to the abscissa axis passing from a point with an ordinate Δ a = 0.318 cm.

Rice. 3.3. Absolute Additive Error Plot

In the resulting graph (see Fig. 3.2) dependences Δ with i= f(L ET), the graph of the multiplicative component Δ m = f(L THIS). The results of calculating the absolute multiplicative error are shown in Table 3.1, and the graph in Figure 3.4.

Rice. 3.4. Plot of the absolute multiplicative error

Proceeding from the fact that the used measuring instrument has a uniform average relative error δ av 12.7% for the entire scale, which is calculated by formula (2.5) in the 2nd section of this work and was used to isolate the additive and multiplicative components of measurement errors in this section work, then the graph of this error will be a horizontal line with an ordinate of 12.7% for the entire range of variation of the linear size L ET.

Let us calculate the relative additive components of the error (δ a i) for each measurement by the measuring instrument, using the obtained value Δ a = 0.318 cm and the dependence of the form:

The results of calculating the relative additive components of the errors (δ a i) are presented in Table 3.2, and the graph in Figure 3.5.

Using the results of calculations of the absolute multiplicative component of the error, which are shown in Table 3.1, we calculate the relative additive components of the error (δ m i) for each measurement by the measuring instrument, using a dependence of the form:

The results of calculating the relative multiplicative components of the errors (δ m i) are presented in Table 3.2, and the graph in Fig. 3.6.

Table 3.2

Results of calculating the relative components of measurement errors

Member number	L this i, cm	δ cf, cm	δ a i, cm	δ m i, cm
	2,5	12,7	12,72	0,0
		12,7	6,36	6,3
	7,5	12,7	4,24	8,5
		12,7	3,18	9,5
	12,5	12,7	2,544	10,2
		12,7	2,12	10,6
	17,5	12,7	1,8	10,9
		12,7	1,6	11,1
	22,5	12,7	1,4	11,3
		12,7	1,3	11,4

Rice. 3.5. Graph of relative additive error

Rice. 3.6. Relative multiplicative error plot

CONCLUSIONS

The completed test allowed:

1) calculate the absolute, relative and reduced errors in the measurement results of the linear size of the structure of the building under construction, the average values of which were respectively:

∆ cf = 1.82 cm, %, .

2) calculate and build graphs of the total absolute and relative errors of the results of measurements of the linear size of the chemical equipment structure, select from them and build graphs of the additive and multiplicative components of the errors;

Any measuring instrument has a static characteristic, i.e. characteristic that functionally connects the output quantity Y with the input quantity X. Typically, the static characteristic is linear. In the absence of errors, the following relation is valid for it

where Y n - nominal static characteristic of the measuring instrument; S n is the nominal sensitivity of the measuring instrument.

The presence of an error in the measuring instrument causes a change in sensitivity ( S n + D S), as well as displacement of the measurement result by the value D a, i.e.

Y= (S n + D S) × X+ D a.

Uncertainty D Y the measurement result will be determined as

D Y=Y– Y n = D S× X + D a.

The first component of the error is multiplicative (D m = D S× X), and the second is additive (D a = D a).

Let us give a definition of additive and multiplicative errors.

Additive called the error absolute value which is invariable over the entire range of the measured value.

The systematic additive error shifts the nominal characteristic in parallel up or down by ± D a (Fig. 5.2).

An example of a systematic additive error is the error from the inaccurate setting of the device to zero, from the contact emf. in a direct current circuit. Additive error is also called zero error.

Multiplicative called the error absolute value which changes in proportion to the measured value.

With a systematic multiplicative error, the real characteristic deviates from the nominal up or down (Figure 5.3).

Examples of systematic multiplicative errors are errors due to a change in the division ratio of a voltage divider, due to a change in the spring stiffness of a measuring mechanism, etc. The multiplicative error is also called the sensitivity error.

In measuring instruments, additive and multiplicative errors, as a rule, are present at the same time. In this case, the resulting error is determined by the sum of the additive and multiplicative errors D = D a + D m = D a + d m × NS, where d m is the relative multiplicative error. Depending on the ratios of the additive (D a) and multiplicative (D m) errors, the accuracy classes of measuring instruments are designated differently. Three typical cases of the ratio of these errors can be distinguished: 1) D a = 0, D m ¹ 0; 2) D a ¹ 0, D m = 0; 3) D a @ D m.

1. By the way of expressing the errors are divided:

On absolute;

Relative;

Given.

Absolute error defined as the difference between the measured and actual values of the measured quantity (formula 4):

The absolute error is expressed in units of the measured quantity.

The absolute error cannot serve as an indicator of accuracy, since it is independent of the measured value. For example, measurement error = 0.5 mm when measuring length = 100 mm corresponds to a sufficiently high measurement accuracy, and at = 1 mm - low.

Relative error is represented as the ratio of the absolute error to the actual value of the measured value. The relative error is found from the ratio (5):

(5)

The relative error is a more accurate characteristic and the most informative, since it makes it possible to compare the results and evaluate the quality of measurements performed at different times, by different means or operators.

However, the relative measurement error cannot be used to normalize the error of measuring instruments, since when the measured value approaches zero, its insignificant changes lead to huge changes.

To eliminate this disadvantage, the concept of reduced error is introduced.

Reduced error Is the ratio of the value of the absolute error to the constant normalizing value (formula 6):

(6)

Either the upper limit of the one-sided scale of the measuring instrument is taken as the normalizing value or measuring range

2. By the nature of the dependence on the measured value errors are divided into additive and multiplicative.

Additive error(zero error) is the error of a measuring instrument that remains constant over the entire measurement range, i.e. the additive error does not depend on the value of the measured value.

Additive, for example, is the error caused by inaccurate zeroing of a dial gauge with a uniform scale.

The multiplicative error(sensitivity error) is the error of a measuring instrument that increases or decreases with an increase in the measured value, i.e. the multiplicative error varies in proportion to the measured value.

The multiplicative, for example, is the error in measuring time intervals by a lagging or rushing clock. This error will increase in absolute value until the owner of the watch sets them correctly according to the exact time signals.

3. By the nature of the manifestation errors are divided into systematic, random and gross (misses).

In the general case, the error of the measurement result includes systematic and random components (formula 7):

where is the systematic component of the total error, is the random component of the total error (the gross error is included in the random component).

Systematic error measurement is called the component of the error of the measurement result, which, when repeated measurements of the same quantity under the same conditions, remains constant or changes regularly, usually progressing.

Systematic errors can be caused by insufficiently accurate implementation of the accepted principle and method of measurements, design flaws of the measuring instrument.

To systematic permanent errors (remaining constant during repeated measurements) can be attributed to the error caused by the temperature deformation of the measured part, and the error of the measuring instrument when the temperature deviates from normal conditions.

An example of a systematic progressive error (regularly changing during repeated measurements) is the error caused by wear of the measuring tip of the measuring instrument during contact measurements.

A distinctive feature of systematic errors is the predictability of their behavior. Since they distort the measurement result, they must be eliminated by introducing corrections or adjusting the device with bringing systematic errors to an acceptable minimum.

Amendment - this is the value of the quantity entered into the uncorrected measurement result in order to eliminate the components of the systematic error. By introducing amendments, as a rule, a systematic constant error of measuring instruments is excluded.

With the introduction of the amendment, the measurement equation will have the form (formula 8):

where is the indication of the measuring instrument; - the value of the measured value; - systematic measurement error; - amendment.

The correction is numerically equal to the value of the systematic error and is opposite to it in sign .

The value of the quantity obtained during the measurement and refined by introducing into it the necessary corrections for the effect of systematic errors are called corrected measurement result.

Systematic errors in the case when they are known and their values in the form of amendments are indicated in the normative and technical documentation (passport) for the measuring instrument, should be taken into account in each of the measurement results.

Systematic constant errors can also be identified (detected) by comparing the measurement results with other, obtained by more accurate methods and means.

In a number of cases, it is possible to get rid of systematic errors in whole or in part during the measurement process, even when they are unknown neither in magnitude nor in sign. For example, when compensating by sign, the measurement is organized in such a way that the systematic error is entered once with one sign, and another time with the opposite one. Next, the arithmetic mean of the two results is taken - while the systematic error is excluded.

By random error measurement is called the component of the error of the measurement result, which, when repeated measurements of the same quantity under the same conditions, changes unexpectedly, randomly.

There are many reasons that cause random errors, for example, distortions of the device elements, fluctuations in the ambient temperature, rounding off of the device readings, changes in the operator's attention, etc.

In the manifestation of these errors, no regularity is observed; they are detected during repeated measurements of the same value in the form of a certain scatter of the results obtained.

Random errors are inevitable, irreparable and always present as a result of the measurement. Unlike systematic errors, random errors cannot be excluded from the measurement result by introducing corrections, but they can be significantly reduced by increasing the number of single measurements. This makes it possible, using the methods of probability theory and mathematical statistics, to refine the result, i.e. bring the value of the measured value closer to the true one.

The random error of the measurement result also includes a slip or gross error.

By mistake (gross error) is the error of a measurement result included in a series of measurements, which for these conditions differs sharply from the rest of the results of this series.

Mistakes, as a rule, arise due to errors or incorrect actions of the operator, incorrect readout of the instrument readings, abrupt short-term changes in conditions during measurements, etc. The moment of the appearance of mistakes is accidental and unknown for the experimenter. With multiple measurements, the set of results obtained may contain several results that contain gross errors.

If misses are detected during the measurement, then the results containing them are discarded as invalid. As a rule, misses are detected based on the analysis of measurement results using various probabilistic criteria.

The division of errors into systematic and random is of great importance in the development of methods for reducing errors, but it is not always easy to implement. Sometimes, depending on the method of performing the same measurement, the error in the result can be both systematic and random.

4. By source of occurrence errors are divided into methodical, subjective and instrumental.

Methodical error (error of the measurement method) - this is a component of the measurement error due to the shortcomings of the theory or method of measurement.

This error arises due to: admitted simplifications during measurements, due to inaccuracy in transferring the size of a quantity from an object to a measuring instrument, errors in data processing, etc.

The methodological components also include the error components due to the limited accuracy of the formulas used to find the measurement result, and the imperfection of the techniques with which the measurement principle is implemented. An example of such an error is the indirect measurement of electrical resistance based on Ohm's law (using an ammeter and a voltmeter). Depending on the connection of the devices, the readings of one or the other contain systematic errors, which leads to an error in the result.

In most cases, methodological errors are systematic in nature, but their accidental manifestation is also possible. For example, if the equations of the measurement method include coefficients depending on the measurement conditions, which vary randomly.

The main feature of methodological errors is the fact that they cannot be indicated in the instrument's passport, but must be assessed by the experimenter himself, i.e. methodological errors do not depend on the manufacturing quality of the measuring instrument.

Subjective error (reading error, personal error) Is the operator-dependent component of the measurement error.

This error is due to the individual characteristics of the operator (inattention, lack or lack of qualifications), the influence of the operator's heat radiation on the measuring instrument.

Such an error is manifested in cases when the reading of indications and the fixation (registration) of the results of observations are carried out either by the operator or automatically; their main reason is inaccuracy, rounding of readings.

Subjective errors cannot be indicated in the passport for the measuring instrument. Therefore, in order to avoid them, it is necessary to follow the rules of operation of measuring instruments, improve the skills of working with measuring equipment and improve the reading devices.

Instrumental error (instrumental, instrumental) Is the component of the measurement error caused by the error of the used measuring instrument.

This error is determined by the imperfection of the measuring instrument, design and technological limitations, the influence of external conditions.

Instrumental error includes the error of the measuring instrument and the error in the interaction of the measuring instrument with the object.

The error in the interaction of a measuring instrument with an object arises due to the fact that the transfer of information is always associated with the selection of some kind of energy from the object. The interaction of a measuring instrument with an object can be different in physical nature: mechanical, electrical, thermal, etc. However, in any case, it is associated with the energy exchange between the object and the measuring instrument, taking place in time and space.

Instrumental errors usually also include interference at the input of a measuring instrument caused by its connection to the object of measurement. For example, when a measuring device is connected to an electrical circuit, the operating mode of this circuit changes.

It is necessary to distinguish between the error of the measuring instrument and the error of the measurement. The measurement error is only a part of the measurement error.

5. According to the conditions of using the measuring instrument errors are divided into basic and additional.

Basic error- the error of the measuring instrument in normal (laboratory) conditions of use, due to the properties of the measuring instrument.

These conditions are established by regulatory and technical documents for the types of measuring instruments or their individual types. Establishing the conditions of use and especially the normal conditions is very important to ensure the uniformity of the metrological characteristics of measuring instruments.

The basic error may include variation error, manifested in the difference in the readings of the measuring instrument at the same point of the measurement range for different directions of approach to this point; calibration error caused by inaccuracies of the exemplary means used in the process of calibrating the measuring instrument; quantization error- rounding operation in digital measuring instruments.

Additional error- the component of the error of the measuring instrument, which arises in addition to the basic error due to the deviation of any of the influencing quantities from its normal value or due to its going beyond the normal range of values.

For example, in operating conditions when installing a measuring instrument on an aircraft, it will have to work when the ambient temperature changes in the range of ± 50 ° C, pressure from 10 2 Pa to 10 4 MPa, supply voltage by 20%, which will cause errors significantly exceeding the main one. ...

The main and additional errors are determined in a static mode, therefore they refer to static errors, which will be considered in the next paragraph.

6. According to the conditions for changing the measured value errors are divided into static and dynamic.

According to the dependence of the absolute error on the values of the measured value, errors are distinguished (Fig. 3.1):

· Additive, independent of the measured value;
· Multiplicative, which are directly proportional to the measured value;
· Nonlinear, having a nonlinear dependence on the measured value.

These types of errors are sometimes also referred to as:

additive ---- zero error;

multiplicative ----- the error of the slope of the characteristic;

nonlinear --------- nonlinearity error.

Rice. 3.1.

Due to the fact that the additive and multiplicative components of the error are characteristic of a measuring instrument, moreover, in the range of measured values, then, based on the given true (actual) value of the linear size of a structural element (17m), let us assume that the used measuring instrument allows measurements in range from 1 m to 100 m, and has the same average relative error for the entire scale, which is calculated by formula (2.5) in the 2nd section of this work. Based on the selected measuring range of the measuring instrument (1m - 100m), take from it, for example, 10 equidistant fixed (reference) values of the linear size of a structural element, including the specified true (actual) value equal to 17 meters. As a result, a number of measured reference values of linear dimensions used by the measuring instrument will have the form: 7; 17; 27; 37; 47; 57; 67; 77; 87; 97 (m).

Using expression (2.5), it is possible to determine the values of the total absolute error for all members of the series, namely:

The calculated values of the total absolute error for all members of the series, taking into account the fulfillment of the rules for rounding the measurement results and measurement errors (given in Appendix 1), are presented in Table 3.1.

Table 3.1.

Results of calculating the total, additive and multiplicative absolute errors

Member number

Using the results of calculations of the total absolute error and a number of measured reference values of linear dimensions, a graph (see Fig. 3.2) of the dependence is built, while the points on which it is constructed are approximated. On the axes of the graph, the initial and final values of the measuring range of the measuring instrument (Len = 1 m and Lek = 100 m) and the maximum value of the total error D s (D ck = - 11.5 m) are indicated.

The resulting graph (Figure 3.2) highlights the additive component (D a) of the total absolute error (D s), which is equal to the total absolute error at the minimum (initial) value of the reference values of linear dimensions (at the beginning of the SI measurement range), i.e. D a = - 0.89 m.

A graph (Figure 3.3) of the dependence of the absolute additive error D a = f (L ET.i) is constructed, which is a straight line parallel to the abscissa axis passing from the point with the ordinate D a = -0.89 m.

Figure 3.2.

Figure 3.3.

On the resulting graph (see Figure 3.2) of the dependence, a graph of the multiplicative component D m = f (L ET) is highlighted, which runs parallel to the graph of the total absolute error, but starts not from a point with coordinates (7; 0.89), but from points with coordinates (7; 0), because , then and On the axes of the graph, the initial and final values of the range of variation of the linear size L ET (Len = 7 m and Lec = 97 m) and the maximum value of the multiplicative error D m (D μ = 11.5 m) are indicated. The results of calculating the absolute multiplicative error are shown in Table 3.1, and the graph in Figure 3.4.

Proceeding from the fact that the used measuring instrument has a uniform average relative error of -12.7% for the entire scale, which is calculated by formula (2.5) in the 2nd section of this work and was used to highlight the additive and multiplicative components of measurement errors in this section of the work , then the graph of this error will be a horizontal line with an ordinate of -10.0% for the entire range of variation of the linear size L ET.

Let us calculate the relative additive components of the error () for each measurement by the measuring instrument, using the obtained value

D a = -0.89 m and dependence of the form:

The results of calculating the relative additive components of the errors () are presented in Table 3.2, and the graph in Figure 3.5.

Figure 3.4.

Table 3.2.

The results of calculations of the relative components of the measurement errors.

Member number

Figure 3.5.

Using the results of calculations of the absolute multiplicative component of the error, which are given in Table 3.1, we calculate the relative additive components of the error () for each measurement by the measuring instrument, using the dependence of the form:

The results of calculating the relative multiplicative components of the errors () are presented in Table 3.2, and the graph in Figure 3.6.