The dynamic characteristics of MIs.

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4.5. The characteristics of MI interaction with an object being tested and with a load:

- the input impedance;

- the output impedance.

The non – informative parameters of the MI output signal.

Here are some explanations about the last characteristic . For example, the material measure of electric oscillation frequency is a generator that produces an electric signal of a stable, precisely known frequency.

The frequency itself is the informative, principal in the given example parameter of the signal produced.

But the same signal may be characterized by the others, non-principal, but auxiliary for this signal parameters, for example, voltage, harmonic factor, etc. These parameters are called non-informative.

For certain Mis those characteristics are chosen, that are sufficient to estimate the measurement errors. Most often these characteristics are used to standardize MIs, and to define the accuracy classes and maximum permissible errors of MIs.

The maximum permissible error of a measurement instrument is the largest (by modulus) error of the MI which allows the instrument to be applicable and permitted for usage.

The accuracy class of a measurement instrument is a generalized characteristic of the MI, which is defined by the maximal permissible intrinsic and complementary errors as well as by other properties of the MI influencing the accuracy, the values of which are defined in standards for different types of MIs.

Though the accuracy class characterizes a complex of the metrological properties of the given MI, it doesn’t determine uniquely the accuracy of measurements made with it. To a great extent, the accuracy of measurements depends on the used method, as well as on conditions it is performed in, etc. That’s why by an incompetent MI usage it is possible to obtain the results containing the big errors even with instruments of the highest accuracy classes.

And vice versa, the competent usage of MIs with not high accuracy classes yields rather precise results.

The maximum permissible errors of measurement instruments used to standardize the accuracy classes are represented in a form of the absolute, relative or fiducial errors according to the character of the error changes in a measurement range as well as to usage conditions and the MI destination.

In Table 4, the different ways of the MI accuracy classes standardization and designation are summarized.

Table 4

The ways of standardization and designation of accuracy classes of MIs

Form of the MI error repre sen-tation	Formula for the maximum permissible error of MI representation	Designation of MI accuracy class	Example
Absol-ute	1) =a 2) =(a+bx) 3) table of error values	Rome numerals Latin alphabet	II,IV B,D
Relati-ve,	1)=100=q 2) (X_m – range of MI, X – indication of MI)	q f/g;(fc;gd)	0,05/0,02
Fiduci-al,	=100=±c if X_f– is a geometrical length of a scale, then:	c	1,5

When the maximum permissible intrinsic MI error is represented in the values of a quantity being measured, i.e., as the absolute error, the following three variants are the most often used:

1. When the absolute error is represented with one value:

=a ,

where a is a constant quantity that is represented in the units of a quantity under measurement. As it is seen from the formula, it is an additive error, i.e., such error that doesn’t depend on the quantity being measured and is constant in the whole range of measurements.

2. When the absolute error is represented as a sum of two components, one of which doesn’t depend on the quantity being measured and other depends on the X (the multiplicative component of the error):

=(a+bx),

where a,b are the constant numbers, that don’t depend on the quantity being measured.

3. When the absolute error is represented as a table of the maximum permissible absolute errors for the different MI indications.

At standardization of the MI accuracy classes with the absolute errors, the Rome numerals (I, II, III, etc.) or capital letters of the Latin alphabet (A, B, C, D, etc.) are used for their designation.

It is significant, that the accuracy class designation with Roman numerals or Latin capital letters contains no numerical information about the permissible errors.

They may become known only thing may be said to compare the similar Mis is that the higher accuracy class is represented with the smaller Roman numeral or with the letter closer to the beginning of the Latin alphabet, e.g., MI of class II is more precise, than the similar MI of class III and MI of class B is more precise than the similar MI of class C.

For many MIs it is convenient to represent the maximum permissible intrinsicerror δ% withtherelative value, usually in percentage.

For example: δ% = 100· Δ/x = ± q ,

where Δ is the maximum permissible intrinsic error; x is a measured quantity; q is some positive number.

Such form is used ,when Δ is represented by the formula:

Δ= ± а .

When Δ is represented with the formula:

Δ= ± (а + b x) ,

the maximum permissible relative intrinsic error (percentage) is then represented by the formula: δ% = ± [c + d ( ½Х_m / x½- 1)] ,

where X_m is the largest (by modulus) limit of the range; c, d are the positive numbers defined as:

с= (b + a / ½ Х_m½) · 100; d = 100 a /½ Х_m½.

The maximal permissible relative intrinsic error, when necessary, is represented by more complicated formulas, diagrams or tables.

At standardization of the MI accuracy classes by the relative error, Arabic numbers from the line:1∙10ⁿ; 1,5∙10ⁿ; 2∙10ⁿ; 2,5∙10ⁿ; 3∙10ⁿ; 4∙10ⁿ;6∙10ⁿ , where n = 1;0;-1;-2, etc. are used for their designation. Values 1,6 and 3 for new models of MIs are not established.

The values q, c and d of the above-mentioned formulas for the relative error are represented with the numerals of this line. Hereby if the accuracy class determined by the relative error is designated with one number, this number is encircled.

For example, a direct current bridge has the accuracy class of 0.1. It means that its maximum permissible relative error

δ%max = 0,1 %.

If necessary it is very easy to calculate the maximum permissible absolute error of a measurement. For example, if a result of a resistance measurement with this bridge R= 500 Ohm, the maximum permissible absolute error Δ= ± 0,5 Ohm, i.e. 0,1 % from 500 Ohm.

Hence:                           R= (500,0 ± 0,5) Ohm

In Table 4 the second formula for the relative error is convenient to be used for standardization of digital measuring instruments accuracy classes. Herewith, the accuracy class is represented with two numbers as the fraction f/g, where f ≥ c, g ≥ d. The values f and g are chosen from the line of numbers given above.

For example, a digital voltmeter with the upper limit of the range 999 V belongs to the 0,05 / 0,02 accuracy class and displays the measurement result 286 V. The maximum permissible intrinsic relative error is calculated as:

            .

The maximum permissible intrinsic absolute error may be calculated as 0,1 % from the measurement result, i.e. from 286 V:

         Δ_max= ± 0,286 V.

For many MIs, e.g. deflectional instruments, the accuracy classes have to be standardized by the fiducial error (in percentage):

                       γ %_max = 100·Δ_max / X_f ,

where γ %_max is the maximum permissible fiducial error; Δ_max is the maximum permissible absolute error; X_f is the fiducial value.

For the general-purpose deflectional instruments the accuracy classes are established as follows: 0,05; 0,1; 0,2; 0,5; 1,0; 1,5; 2,5; 4,0.

This number of accuracy class represents the maximum permissible intrinsic fiducial error of an instrument expressed in percentage.

Let us consider the illustrations for the deflectional instruments with different fiducial values.

1. Voltmeter has a scale of (0…300) V. The accuracy class is 4,0. Calculate the permissible absolute error of a voltage measurement with this instrument.

The accuracy class designation of 4,0 means that for this instrument the permissible intrinsic fiducial error doesn't exceed 4 %. To calculate the permissible absolute error we need to use the formula for the fiducial error γ %, where γ % = 4%, X_f =300V. Thus, we obtain: Δmax= ± 12 V.

The same result may be obtained by simply taking 4% from the fiducial value, i.e. from 300 V. It could be understood in the following way: if the instrument indicates, for example 100 V, then the actual value of the measured voltage is not less than 88 V; i.e. (100 V - 12 V), and not more than 112 V, i.e. (100 V + 12 V).

The same permissible absolute error Δ_max= ± 12 V is possible for any indications of the voltmeter.

This example shows why the accuracy class of a deflectional instrument cannot be standardized by the relative error. The matter is that for different indications the relative error is changed while the absolute error remains constant. For example, for measurement of 300 V voltage the permissible relative error will be 4 %, for measurement of 100 V - 12 % and at an attempt to measure the voltage of 12 V - 100 %.

The last result of an incompetent measurement proves once more the well-known rule of a deflectional instrument measurement range choice.

A measurement range of a deflectional instrument has to be chosen in such a way that the result would be read as close to the scale end as possible, preferably in the last third of the scale.Then the relative errors will be diminished.

2. An ammeter has a scale (-20…0…+60) А. The accuracy class is 2.5. Calculate the permissible absolute error of the current measurements with this instrument.

As it is known for such scale, the fiducial value equals to the sum

of the ultimate scale marks modules, i.e. 80 A. So, the permissible absolute error may be calculated as 2,5 % from 80A, i.e. Δ_max= ± 2 А. If, for example, this ammeter indicates 50 A, it means that the measured current is:

I = (50 ± 2) A,

It means that the actual current value is not less than 48 A, and not more than 52 A.

3. A kiloohmmeter has a scale that is shown in Fig. 10.

Fig. 10. Scale of kiloohmmeter

As it is seen from Fig. 10, the scale is very nonuniform and has on one of the ends the mark ∞. For these scales the biggest scale angle or, that is more convenient in practice, the geometrical scale length l_maxis considered as the fiducial value. This value is specified in the data sheet of the instrument. Let it be, for example, l_max = 150mm. That is for this instrument X_f = 150 mm.

Pay your attention to the fact, that according to such standartization of X_f, the designation of the accuracy class is accompanied by an “angle”(or by a tick) 4.0 .

Calculate the permissible absolute error of a resistance measurement, when the instrument indicates

R = 0,3 kOhm.

The permissible absolute error may be calculated like in the previous example, i.e.as 4 % from the fiducial value, i.e. from 150 mm. We will gain: Δmax= ± 6 mm. But the result of the resistance measurement cannot be written in the form of:

                           R = 0,3 kOhm ± 6 mm.

It is necessary to convert those millimeters to kOhms. For this purpose we have to sign from the reading on the scale (0,3 kOhm) both to the left and right sides on 6 mm and estimate approximately those two results (approximately 0,28 and 0,34 kOhm).

Thus,the correct form of the result writing is:

R = (0.30   ) kΩ

Pay your attention, that the tolerance due to the scale nonuniformity is dissymmetrical.

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