Short theoretical information about estimation of uncertainty of measurement results of physical quantities.

по лабораторной работе №28

 

 

“Измерение удельного заряда электрона методом магнетрона”

 

 

Студент: Леонов Александр

Группа: Р-122 КТ

Дата: 23.10.01

 

 

г.Краснотурьинск

2001г.

I. Расчетная формула для определения удельного заряда электрона с пояснениями смысла величин, входящих в нее.

где U_a=5,25 В – разность потенциалов между катодом и анодом,

L – длинна соленоида,

D – диаметр соленоида,

R_a – радиус анода,

μ₀= 4* 10^-7Гн/м – магнитная постоянная,

I_c,kp – критический ток в соленоиде,

N – число витков соленоида.

 

II. Средства измерений и их характеристики.

Наименование средства измерения и его номер	Предел измерений или номинальное значение	Цена деления шкалы	Класс точности	Предел основной погрешности θ;осн.
Вольтметр	3В	0,05		0,025
Микроамперметр				0,5
Амперметр		0,1		0,05

Магнетрон:

А) Соленоид – диаметр D=80 мм, длинна L=120 мм, число витков N=200; Δ_D=1мм, Δ_L=1мм;

Б) Диод – радиус анодаR_a=3,36мм, Δ_R=0,01мм.

Погрешности:

А) ΔU_a= θ; осн_.=в (задается в таблице к установке);

Б) ΔI_c,кр=А (задается в таблице к установке).

 

III. Схема электрической цепи.

 

IV. Результаты измерений (в форме табл. 1).

 

Таблица 1

Зависимость анодного тока от тока в соленоиде

Ic, А		0,1	0,2	0,3	0,4	0,5	0,6	0,7	0,8	0,9
Ia, мкА											12,5
1,1	1,2	1,3	1,35	1,4	1,45	1,5	1,55	1,6	1,65	1,7	1,8
	11,5	10,5	9,5				7,5			6,5
1,9		2,1	2,2	2,3	2,4	2,5	2,6	2,7	2,8	2,9
	4,5		3,5		2,5			1,5

 

V. Построение графика I_a=f(I_c).

 

VI. Определение критического тока I_{c, кр} в соленоиде по графику, построенному по данным табл. 2.(13

 

Таблица 2

|ΔIa|, мкА										0,5	0,5
ΔIc, А	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1
|ΔIa|/ ΔIc, *10-6
<ΔIc>	0,05	0,15	0,25	0,35	0,45	0,55	0,65	0,75	0,85	0,95	1,05
0,5			0,5			0,5	0,5		0,5	0,5
0,1	0,1	0,05	0,05	0,05	0,05	0,05	0,05	0,05	0,05	0,1	0,1
1,15	1,25	1,325	1,375	1,425	1,475	1,525	1,575	1,625	1,675	1,75	1,85
0,5	0,5	0,5	0,5	0,5	0,5		0,5	0,5
0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1	0,1
1,95	2,05	2,15	2,25	2,35	2,45	2,55	2,65	2,75	2,85	2,95

 

VII. Удельный заряд электрона

6,8868772*10¹⁰Кл/кг

VIII. Оценка границ погрешностей результата измерения:

IX. Окончательный результат |e|/m=(6,8868 ±0,0007)*10¹⁰ Кл/кг

 

X. Выводы: В результате проделанной работы я познакомился с измерением заряда электрона методом магнетрона, получил зависимость анодного тока от тока в соленоиде. В результате получил конкретные значения, которые сравнил с теоретическими. Между ними появились расхождения, т.к. погрешность имеют все приборы (особенно большую погрешность имеет вольтметр), в проводниках имеется активное сопротивление, диаметр и длинна соленоида рассчитаны не точно и многое другое, а так же не совершенность метода. По графику прослеживается зависимость анодного тока от тока в соленоиде.

Short theoretical information about estimation of uncertainty of measurement results of physical quantities.

 

Physical quantity - is defined as property which is common in a qualitative sense for many physical objects (physical systems, their states and processes occurring in them), but in a quantitative sense individual for each object. The size of physical quantity is named as quantitative content of properties in this object, corresponding to the notion of a physical quantity.

Unit of physical quantity - physical quantity of a certain size, adopted in agreement for the quantitative mapping of homogeneous with it values. Various units of the same size vary in size.

The size of units of physical quantities - quantitative characterization of the unit defined by the number of values ​​in the unit of measurement. The size of basic units is established arbitrarily and independently of one another by definitions. The size of the derivative unit is determined by the type of depence between the quantities and sizes of units and the established from equations which define this unit from the basic or other derived units.

The dimension of the of PV - an expression showing connection of the value of physical quantities are the basis in system of units. The dimension is the qualitative characteristic of FV can be written as a product of symbol (dimensions) corresponding to the basic values​​, raised to some degrees, called the dimension indices. The numerical value of the dimension indices determined from the connection equation of derivatives and the main physiacl values. Values, in which all the main values ​​are zero degree, called dimensionless.

Measurements, as experimental processes are varied.This is explained by many experimental quantities the different character of the measurement values, the different requirements of accuracy, etc.

The purpose of measurement is to determine the value of the measured quantity, i.e. values ​​of a certain size, which must be measured. Therefore, the measurement starts the corresponding specification of the measured value, the measurement method and measurement procedure.

Most widespread classification of the types of measurements, depending on the method of processing the experimental data. In accordance with the classification of the measurement can be divided into direct, indirect, joint and cumulative.

Direct measurement - a measurement in which the required value of a physical quantity is directly from experimental data by comparing the measured value with the standards (measurement of the length by ruler, measure the voltage by voltmeter).

Indirect measurement - measurement, in which the required value of is based on the known dependence between this value and the values ​​he chooses to direct measurements (for example, we find the resistance of resistor on the basis of Ohm's law by substituting the current and voltage values ​​obtained by direct measurements.)

Joint dimension - the simultaneous measurement of several variables which required values variables are solution of the system consisting of a result of direct measurements of various combinations of these variables.

The combined measurement - simultaneous measurement of multiple variables of the same name, in which the required values of variables are solution of the system consisting of a result of direct measurements of various combinations of these variables.

Currently, evaluation of the quality of PV measurement performed using the term "measurement uncertainty". The word "uncertainty" means doubt and, thus, in its broadest sense, the "measurement uncertainty" means doubt about the reliability of the measurement result. In the narrow sense of the uncertainty of measurement - is a parameter related to the measurement result, which characterizes the scattering of the values ​​that could reasonably be assigned to the measurand.

In other words, the uncertainty of measurement is understood as incomplete knowledge about the importance of the measured value, and to quantify the incompleteness of the use of the probability distribution of possible law (reasonably assigned) values ​​of the measured value. The parameter of this distribution (also called - uncertainty) quantifies the accuracy of the measurement result.

The conceptual basis of the replacement of the term "error" in the measurement result, the term "uncertainty" is the philosophical premise of agnosticism about what the true meaning unknowable, and the error based on the use of the true value of the measured value, looses meaning.
Thus, at present there was a departure from the developed methods for estimating the accuracy of the measurement results based on the distribution of errors in the random and systematic component.

In the classical approach to the estimation the accuracy of measurements was taken as a postulate that:

a) There is the true value of the measurand;
b) the true value cannot be determined;
c) the true value - the value is a nonrandom.

 

In the current international and Ukrainian regulations [ 1, 2, 3 ] provides a new concept of processing the results of measurements of PV. The main thing in it is:
- Not using the terms "accuracy" and "the true meaning of the quantity being measured" in favor of the concepts of "uncertainty" and "estimated value of the measurand"
- The transition from classification errors on the nature of their manifestation in the random and systematic division to another according to the method of estimation of measurement uncertainty (Type A - the methods of mathematical statistics and the type B - using the characteristics of the measured object, and SIT) is not identical to the previous classification.
In evaluating the uncertainty of type B should be considered as the main distribution laws and the additional error of SIT.
The following types of uncertainty: standard, enhanced, and the total standard uncertainty.
The standard uncertainty (H): the uncertainty of measurement results, expressed as standard deviation (СКО), it is the primary quantitative expression of measurement uncertainty
The total standard uncertainty (Hc): standard uncertainty of measurement results to the square root sum of terms, and terms are the variances or covariances of these other quantities weighted according to how the measurement result varies with these quantities, it is the primary quantitative expression of measurement uncertainty, with where the result is determined by the values ​​of other variables
Expanded uncertainty (Hp) value that defines an interval around the measurement result within which, as you might expect, there is a big part of the distribution of values ​​that reasonably could be attributed to the measurand.
In general, the measured value of Y can be written as:

Y = ƒ;(x 1,…, xm),

(1)

where x1,..., xm - input values ​​(or other directly measured quantities influencing on the measurement);
T - the number of these units;
f - kind of functional dependence.
There are two types of calculations the standard uncertainty:
- calculation by type A - by statistical analysis of measurement results;

- calculation by type B - using other methods.
Estimation of uncertainty of measurement results by type A and by type B is produced as the direct and the indirect methods of measurement.

Calculation of uncertainty according to Type A

Initial data for calculation of HA by type A is a statistical number of measured input variables: {xq} (where q = 1,..., m).
Standard uncertainty of measurement of a single q-th input variable HAq calculated by the formula:

,

(2)

Where - average arithmetic of measurements q-th input variable; n - number of measurements of the input variable.

The standard uncertainty of HA (xq) measurement range of the input variable {xq} (where q = 1,..., m) at which the result is determined as the arithmetic mean is calculated by the formula:

.

(3)

The total standard uncertainty (HcA) in the case of uncorrelated estimates of indirect measurements of ejection fraction ((x ₁,…, х_m) is defined as:

.

(4)

Where - sensitivity coefficient.

In the case of direct measurements of the total standard uncertainty is given by:

.

 

In evaluating the uncertainty of the result of indirect measurements of PV using the concept of correlation and covariance of random variables.
Correlation and covariance - a measure of linear dependence of two variables. Correlation and covariance are one and the same sense as these terms indicate whether there is a linear relationship between two random variables, and can be regarded as a "two-dimensional dispersion." However, unlike the correlation coefficient, which varies from -1 to 1, the covariance is not invariant under the scale, ie scale of the random variables.
The sign of the covariance indicates the type of linear relationship between these quantities, if it is greater than zero - this means a direct connection (with an increase in one quantity increases, and other), the covariance is less than zero indicates the feedback. When the covariance is equal to 0, the linear relationship between the variables is missing.
In the case of correlated estimates of x1,..., xm combined standard uncertainty according to Type A is calculated by the formula:

,

(5)

Where - correlation coefficient

The correlation coefficient, which is a measure of the relative mutual dependence of two random variables is equal to the ratio of their covariance to the positive square root of the product of their variances.

,

Where і, j = 1, 2, …, n; k, q = 1, 2, …, m.

To calculate the correlation coefficient r(x_k, x_q) using a matched pair of measurements (x_kd, x_qd) (where d = 1,..., n_kq; n_kq --the number of matched pairs of measurements).

Calculation of uncertainty of type B

The initial data used to calculate HB:
- The data prior to measuring the quantities in the equation of measurement data on the distribution of probabilities;
- Data based on the experience of the researcher or general knowledge about the behavior and properties of the instruments and materials;
- The uncertainty of the constants and reference data;
- Data verification, calibration, information about the device manufacturer, etc.

To calculate the standard uncertainty of type B - H _B (x_q), influencing factors should be considered:

1) The technical characteristics of the device (the limit of the basic relative error limit of the additional error of the device, the resistive component of the input resistance of the device),
2) The scheme of measurement (the output voltage source with internal resistance, the limit of the measurement device),
3) the measurement conditions (ambient temperature, humidity, pressure, etc.).
Also, to calculate the standard uncertainty must take into account the distribution function (rectangular, trapezoidal, triangular), the main and additional errors.
For example the formula for calculating the uncertainty of measurement results with a voltmeter the voltage is as follows:

,

(6)

Where l = 1, 2_, 3 …. L, a – the main error of the voltmeter, ξ – factors influencing on the measurement result.

The total standard uncertainty of voltage measurement result is as follows:

,

(7)

Where - uncertainty caused by the basic error of the voltmeter;

- uncertainty caused by temperature changes;

- standard uncertainty caused by shunting the output impedance of the object input resistance of the SIT.

Standard uncertainty caused by the basic error of the voltmeter, calculated according to the type B — H _аB (U), determined by the formulas:
- For the rectangular (uniform) distribution function

where a - the main error of the SIT in terms of the accuracy class of instrument. For other distribution functions used by other forms provided in Appendix A.
The uncertainty of measurement results due to the shunt resistance of the source-impedance voltmeter R _вх. At the same time

,

(8)

where U_V – the voltmeter results; – voltage on the load.

Moreover, it is necessary to take into account uncertainty in the input impedance H _{R вх} uncertainty of load resistance H _{R Н}Also it is necessary to determine the correction (ρ;):

,

(9)

Measurement uncertainty introduced by the amendment (ρ) is given by:

,

(10)

 

Where the sensitivity coefficients , ;

– uncertainty of the load resistance and H _{R вх} – uncertainty in the input resistance. Since the value of load resistance and input impedance are known about and is, within certain limits [ R _min; R _max], [ R _вхmin; R _вхmax], considering all the values ​​of resistors within the marked boundaries equally, we estimate their uncertainty as:

To estimate the uncertainty Н_Т⁰ _В, due to deviation from the normal temperature Т_норм = 20˚С assumed that the limit of the instrument additional error caused by deviation of the ambient temperature (Т_окр.ср)_. within the normal operating temperature of the area, which will not exceed the limit of basic error for every 10 ˚ C temperature change is:

.

(11)

For digital equipment necessary to calculate the uncertainty of the quantization Н_{кв. В} (U), defined by the boundaries of the quantization error () taking into account a uniform distribution law:

.

(12)

In the above example suggests that a uniform distribution of the probability of errors in the SIT, the parameters of the object.

The total standard uncertainty Н_cв with indirect measurements in the case of uncorrelated estimates x1,…, хm is calculated by formula:

,

(13)

where - standard uncertainty of measurement results of q-th input value.

The resulting standard uncertainty is determined by the formula:

(14)

Expanded uncertainty of Hp given by multiplying the resulting standard uncertainty of measurement result in the coverafe factor (k).

Нр = k ∙Нå,

(15)

In general, the coverage factor k is selected in accordance with the expression

(16)

where - quantile of the Student distribution with effective degrees of freedom v_eff and the confidence probability (confidence level), p. The values ​​of the coefficient are given in Appendix B.

The freedom degree - a characteristic of PV. The number of degrees of freedom determines the minimum number of independent variables required for a complete description of the process under investigation. Also, the number of degrees of freedom equal to the total number of independent numbers.
The effective number of degrees of freedom for the indirect measurement of multiple determined by the formula:

,

(17)

where - number of degrees of freedom during the estimation of q-th input value; = n_q –1 – during the calculation of uncertainty of type A.

For direct multiple measurements n _eff = n – 1

In many practical cases, when calculating the uncertainty of measurement results and speculate about a normal distribution of possible values ​​of the measured value and assume:

k = 2 if р ≈ 0,95 and k = 3 if р ≈ 0,99.

With the assumption of uniformity of the distribution law considered:

k = 1,65 if р ≈ 0,95 and k = 1,71 if р ≈ 0,99.

When you write the measurement result is the actual value of the measured value (y), the resulting total uncertainty (Н_∑(y)), expanded uncertainty (Hp), the effective number of freedom degrees () and confidence level (р _д).

Laboratory work №8

Estimation of uncertainty of measurement results of physical values

Part 1. Estimation of uncertainty of the results of direct measurements of the voltage of information signals.

1. Purpose of work

1.1. To study the basic kinds and types of uncertainty of measurement results of physical quantities, a methodology for assessing the uncertainty of type A and type B.
1.2. Experiments to evaluate the uncertainty of the results of direct measurements on the result of statistical analysis of the type A and a priori information on the characteristics of the SIT and the measurement of objects of type B.
1.3. Provide a comparative analysis of the estimates of uncertainty of measurement results by type A to type B.

2. Key positions

2.1 Uncertainty of measurement - parameter related to the results of measurements and characterizes the scattering of the values ​​that could reasonably be assigned to the measurand.
The following types of uncertainty:
- standard,
- total standard,
- extended.
2.2 There are two types of calculations the standard uncertainty:
- Calculation of the type A - by statistical analysis of measurement results;
- Calculation of the type B - using a priori information about the measured input quantity and characteristics of the SIT.
In general, the measured value of Y can be represented as (1).

2.3 Calculation of uncertainties of type A. The initial data for calculation of H _А were measured by a number of statistical input variable: x_q. After that it is necessary to calculate standard uncertainty of single measurement of q-th input variable H _Aq (2)
and standard uncertainty H _A (x_q) measurement range of the input variable x_qі, where i = 1, 2,..., n (3).

2.4 During the estimation of uncertainty of measurement results of type B using such formulas (6), (7), (8), (9), (10), (11).

The standard uncertainty of H B (a, ξ) is determined by the formulas for the distribution functions (rectangular, trapezoidal, triangular), the main and additional errors (Appendix A), where a - is determined by the accuracy class instrument.

2.5 The resulting standard uncertainty is determined by the formula (14).

2.6 The effective number of degrees of freedom is determined by - n eff = n - 1.

2.7 Expanded uncertainty (H p) obtained by multiplying the resulting standard uncertainty of measurement results on the enrollment ratio (k) (15).