Accuracy of measuring instruments and measurements. Measurement accuracy

Error is the deviation of the result of measuring a physical quantity (for example: pressure) from the true value of the measured quantity. Error arises as a result of imperfection of the method or technology. measuring instruments, insufficient consideration of the influence of external conditions on the measurement process, the specific nature of the measured quantities themselves and other factors.

The accuracy of the measurements is characterized by the closeness of their results to the true value of the measured quantities. There is a concept of absolute and relative measurement error.

The absolute measurement error is the difference between the measurement result and the actual value of the measured quantity:

DX=Q-X,(6.16)

The absolute error is expressed in units of the measured value (kgf/cm2, etc.)

The relative measurement error characterizes the quality of the measurement results and is defined as the ratio of the absolute error DX to the actual value of the quantity:

d X=DX/ X , (6.17)

Relative error is usually expressed as a percentage.

Depending on the reasons leading to measurement error, there are systematic And random errors.

Systematic measurement errors include errors that, during repeated measurements under the same conditions, manifest themselves in the same way, i.e., they remain constant or their values ​​change according to a certain law. Such measurement errors are determined quite accurately.

Random errors are errors whose values ​​are measured during repeated measurements of a physical quantity, performed in the same way.

The error of instruments is assessed as a result of their verification, i.e., a set of actions (measures) aimed at comparing instrument readings with the actual value of the measured value. When checking working instruments, the actual value of the measured quantity is taken to be the value of standard measures or readings of standard instruments. When assessing the error of standard measuring instruments, the value of the standard measures or the readings of the standard instruments are taken as the actual value of the measurement of the quantity.

The main error is the error inherent in the measuring instrument under normal conditions (atmospheric pressure, Тair = 20 degrees, humidity 50-80%).

Additional error is the error caused by measuring one of the influencing quantities beyond the limits normal conditions. (for example temperature, average measurement)

The concept of accuracy classes. The accuracy class is a generalized characteristic of measuring instruments, defined by the limits of permissible basic and additional errors, as well as other properties of these instruments that may affect their accuracy. The accuracy class is expressed by a number that coincides with the value of the permissible error.

A standard pressure gauge (sensor) of accuracy class 0.4 has an acceptable error = 0.4% of the measurement limit, i.e. the error of a standard pressure gauge with a measurement limit of 30 MPa should not exceed +-0.12 MPa.

Accuracy classes of pressure measuring instruments: 0.16; 0.25; 0.4; 0.6; 1.0; 1.5; 2.5.

Sensitivity devices is called the ratio of the movement of its pointer D n (arrow direction) to the change in the value of the measured quantity that caused this movement. Thus, the higher the accuracy of the device, the greater the sensitivity, as a rule.

Main characteristics measuring instruments are determined in the process of special tests, including calibration, during which the calibration characteristic of the device is determined i.e. the relationship between its readings and the values ​​of the measured quantity. The calibration characteristic is compiled in the form of graphs, formulas or tables.

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§ 32. ACCURACY AND ERROR OF MEASUREMENT.

No measurement can be made absolutely accurately. There is always some difference between the measured value of a quantity and its actual value, which is called measurement error. The smaller the measurement error, the naturally higher the measurement accuracy.

Measurement accuracy characterizes the error that is inevitable when working with the most precise measuring instrument or a device of a certain type. The measurement accuracy is influenced by the material properties of the measuring tool and the design of the tool. Accuracy of measurement can only be achieved if the measurement is carried out according to the rules.

The main reasons that reduce measurement accuracy may be:

1) unsatisfactory condition of the tool: damaged edges, dirt, incorrect position of the zero mark, malfunction;

2) careless handling of the tool (impacts, heat, etc.);

3) inaccuracy of installation of the tool or the part being measured relative to the tool;

4) the temperature difference at which the measurement is made (the normal temperature at which the measurement should be made is 20°);

5) poor knowledge of the device or inability to use a measuring instrument. Incorrect choice of measurement tool.

The degree of measurement accuracy of any device depends on its care, as well as its correct use.

Increased measurement accuracy can be achieved by repeating measurements and then determining the arithmetic mean obtained as a result of several measurements.

When starting to measure, you need to have a good knowledge of the measuring instruments, the rules for handling the instrument and master the techniques for using it.

In power supply systems, current is measured ( I), voltage (U), active and reactive power (P,), electricity, active, reactive and impedance (P, Q), frequency ( f), power factor (cosφ); when supplying energy, measure temperature (Ө), pressure ( R

In operating conditions, methods are usually used direct assessment for measuring electrical quantities and zero - for non-electrical quantities.

Electrical quantities are measured by electrical measuring instruments.

Electrical measuring instrument is a device designed to measure an electrical quantity, such as voltage, current, resistance, power, etc.

According to the principle of operation and design features, devices are: magnetoelectric, electromagnetic, electrodynamic, ferrodynamic, induction, vibration and others. Electrical measuring instruments are also classified according to the degree of protection of the measuring mechanism from the influence of external magnetic and electric fields on the accuracy of its readings, by the method of creating a counteracting moment, by the nature of the scale, by the design of the reading device, by the position of the zero mark on the scale and other characteristics.

On the scale of electrical measuring instruments are marked symbols, defining the device system, its technical characteristics.

Electrical energy generated by generators or consumed by consumers is measured by meters.

To measure alternating current electrical energy, meters with an induction system measuring mechanism and electronic ones are mainly used. The deviation of the measurement result from the true value of the measured value is called measurement error.

Measurement accuracy- quality of measurement, reflecting the closeness of its results to the true value of the measured value. High measurement accuracy corresponds to low error.

Meter error- the difference between the instrument readings and the true value of the measured value.

Measurement result- the value of a quantity found by measuring it.

With a single measurement, the instrument reading is the result of the measurement, and with multiple measurements, the measurement result is found by statistically processing the results of each observation. According to the accuracy of the measurement results, they are divided into three types: full-time (precision), the result of which must have a minimum error; control and verification tests, the error of which should not exceed a certain specified value; technical, the result of which contains an error determined by the error of the measuring device. As a rule, accurate and control measurements require multiple observations.

According to the method of expression, the errors of measuring instruments are divided into absolute, relative and reduced.

Absolute errorΔA is the difference between the reading of instrument A and the actual value of the measured quantity A.

Relative error- the ratio of the absolute error ΔA to the value of the measured quantity A, expressed as a percentage:

.

Reduced error(in percent) - the ratio of the absolute error of the aircraft to the normalizing value:

.

For instruments with a zero mark at the edge or outside the scale, the standard value is equal to the end value of the measuring range. For instruments with a double-sided scale, that is, with scale marks located on both sides of zero, it is equal to arithmetic sum final values ​​of the measuring range. For instruments with a logarithmic or hyperbolic scale, the normalizing value is equal to the length of the entire scale.

Table 1. Accuracy classes* of measuring instruments

Instruments for measuring electrical quantities must satisfy the following basic requirements (PUE):

1) the accuracy class of measuring instruments must be no worse than 2.5;

2) accuracy classes of measuring shunts, additional resistors, transformers and converters must be no worse than those given in table. 1.;

3) the measurement limits of instruments must be selected taking into account the possible largest long-term deviations of the measured values ​​from the nominal values.

Accounting for active electrical energy should ensure the determination of the amount of energy: generated by ES generators; consumed per s. n. and economic needs (separately) of ES and PS; supplied to consumers via lines extending from the ES buses directly to consumers; transferred to or received from other energy systems; released to consumers from the electrical network. In addition, accounting for active electrical energy should provide the ability to: determine the flow of electrical energy into electrical networks of different voltage classes of the power system; compiling balances of electrical energy for self-supporting units of the energy system; monitoring compliance by consumers with their specified consumption regimes and balance of electrical energy.

Accounting for reactive electrical energy should provide the ability to determine the amount of reactive electrical energy received by the consumer from the power supply organization or transferred to it only if these data are used to make calculations or monitor compliance with the specified operating mode of compensating devices.

Current measurement must be carried out in circuits of all voltages where it is necessary for the systematic control of a technological process or equipment.

Direct current measurement in circuits: DC generators and power converters; AB, chargers, sub-chargers and discharge devices; excitation of SG, SC, as well as electric motors with controlled excitation.

DC ammeters must have double-sided scales if the direction of current can be reversed.

In three-phase current circuits, as a rule, the current of one phase should be measured.

The current measurement of each phase must be carried out:

for TG 12 MW or more; for overhead lines with phase-by-phase control, lines with longitudinal compensation and lines for which the possibility of long-term operation in open-phase mode is provided; in justified cases, it may be possible to measure the current of each phase of an overhead line of 330 kV and above with three-phase control; for electric arc furnaces.

Voltage measurement should be done:

1. On sections of DC and AC busbars, which can operate separately. It is allowed to install one device with switching to several measurement points. At a substation, voltage can be measured only on the LV side, if the installation of a VT on the HV side is not required for other purposes.

2. In the circuits of direct and alternating current generators, SCs, as well as in some cases in the circuits of units special purpose.

When automatically starting generators or other units, installing devices for continuous voltage measurement on them is not necessary.

3. In SM excitation circuits from 1 MW or more.

4. In the circuits of power converters, batteries, chargers and rechargers.

5. In circuits of arc suppression coils.

In three-phase networks, as a rule, one phase-to-phase voltage is measured. In networks above 1 kV with an effectively grounded neutral, it is allowed to measure three phase-to-phase voltages to monitor the health of voltage circuits with one device (with switching).

The values ​​of one phase-to-phase voltage of busbars of 110 kV and higher (or voltage deviation from the specified value) of electric power plants and substations, based on the voltage of which the power system mode is maintained, must be recorded.

Insulation monitoring. In AC networks above 1 kV with an isolated neutral or grounded through an arc suppression reactor, in AC networks up to 1 kV with an isolated neutral and in DC networks with isolated poles or with an isolated midpoint, as a rule, an automatic insulation monitoring must be carried out. to a signal when the insulation resistance of one of the phases (or poles) decreases below a specified value, followed by monitoring the voltage asymmetry using an indicating device (with switching). It is allowed to monitor insulation by periodic voltage measurements in order to visually monitor voltage asymmetry.

Power measurement:

1. Active and reactive power generators.

When installing panel indicating devices on a TG of 100 MW or more, their accuracy class must be at least 1.0.

ES 200 MW or more - total active power.

2. Capacitor banks of 25 Mvar or more and SC reactive power.

3. Transformers and lines feeding the village. n. b kV and above ES, active power.

4. Step-up two-winding transformers ES - active and reactive. In circuits of step-up three-winding transformers (or autotransformers using a LV winding), the measurement of active and reactive power must be carried out on the MV and LV sides. for a transformer operating in a block with a generator, the power measurement on the NI side should be carried out in the generator circuit.

5. Step-down transformers 220 kV and above - active and reactive, 110-150 kV - active power.

In circuits of step-down two-winding transformers, power measurements should be made from the LV side, and in circuits of step-down three-winding transformers - from the MV and LV sides.

At 110-220 kV substations without switches on the overhead power side, power measurements may not be performed.

6. Lines 110 kV and above with two-way power supply, as well as bypass switches - active and reactive power.

7. On other elements of the substation, where periodic monitoring of network modes requires measurements of active and reactive power flows, it should be possible to connect portable monitoring devices.

registration must be carried out: the active power of the TG is 60 MW or more; total power of the power plant (200 MW or more).

Frequency measurement:

1. On each section of the generator voltage busbars.

2. At each TG of a block power plant or nuclear power plant.

3. On each system (section) of HV ES buses.

4. At the nodes of possible division of the power system into non-synchronously operating parts.

Registration of the frequency or its deviation from the specified value should be carried out: on power plants of 200 MW or more; at power plants of 6 MW or more, operating in isolation.

The absolute error of recording frequency meters on ES involved in power regulation should be no more than 0.1 Hz.

Synchronization measurements. For measurements with precise (manual or semi-automatic) synchronization, the following instruments should be provided: two voltmeters (or double voltmeter); two frequency counters (or double frequency counter); synchroscope.

Registration of electrical quantities in emergency modes. For automatic registration of emergency processes in the electrical part of power systems, automatic oscilloscopes must be provided. The placement of automatic oscilloscopes at objects, as well as the selection of electrical parameters recorded by them, are carried out according to the instructions of the PUE.

To determine the location of faults on overhead lines of 110 kV and higher with a length of more than 20 km, fixing devices must be provided.

Table 2. Characteristics of measuring instruments

Designation	Device type		Conversion	How to use	Note
	Magnetoelectric (M) Logometer (M)				WITH- constant Coil currents
	Electromagnetic (E) Logometer (E)				Coil currents
	Electrodynamic (D) Logometer (D)				Coil currents Fixed coil current
	Ferrodians- chesical (D) Logometer (D)				Fixed coil current
	Induction (I) Logometer (I)				N - disk revolutions
	Electrostatic chesical (C) Thermal (T) Rectifier (V)

Modern industrial enterprises and housing and communal services are characterized by consumption various types energy: electricity, heat, gas, compressed air, etc. to monitor energy consumption, it is necessary to measure and record electrical and non-electrical quantities for the purpose of further processing of information.

In power supply, current is measured ( I), voltage (U), active and reactive power (P, Q), electricity (W), active, reactive and impedance (R, X, Z), frequency ( f), power factor (cosφ); in energy supply - temperature (Ө), pressure ( R), energy consumption (G), thermal energy (E), movement (X), etc.

The range of instruments used in power supply for measuring electrical and non-electrical quantities is very diverse both in terms of measurement methods and in the complexity of the converters. Along with the direct assessment method, zero and differential methods, increasing accuracy.

Below is given a brief description of measuring instruments according to the principle of operation.

Magnetoelectric devices have high sensitivity, low current consumption, poor overload capacity, high measurement accuracy. Ammeters and voltmeters have linear scales, and are often used as exemplary instruments; they have low sensitivity to external magnetic fields.

Electromagnetic devices They have low sensitivity, significant current consumption, good overload capacity, and low measurement accuracy. The scales are not linear and are linearized in the upper part by a special mechanism. They are often used as panel technical devices, simple and reliable in operation; sensitive to external magnetic fields.

Electrodynamic and ferrodynamic devices have low sensitivity, high current consumption, sensitivity to overloads, and high accuracy. Ammeters and voltmeters have nonlinear scales. An important positive feature is the same readings on direct and alternating currents, which allows you to check them on direct current. More often they are used as laboratory instruments.

Induction system devices characterized by low sensitivity, significant current consumption, and insensitivity to overloads. They primarily serve as AC energy meters. Such devices are produced in single-, two- and three-element versions for operation in single-phase, three-phase three-wire, three-phase four-wire circuits. Current and voltage transformers are used to extend the limits.

Electrostatic devices They have low sensitivity, but are sensitive to overloads and are used to measure voltage on direct and alternating currents. To extend the limits, capacitive and resistive dividers are used.

Thermoelectric devices characterized by low sensitivity, high current consumption, low overload capacity, low accuracy and non-linearity of the scale. However, their readings do not depend on the shape of the current over a wide frequency range. To expand the limits of ammeters, high-frequency current transformers are used.

Rectifier devices characterized by high sensitivity, low current consumption, low overload capacity, and linear scale. The instrument readings depend on the shape of the current. They are used as ammeters and voltmeters.

Digital electronic measuring instruments convert the analog input signal into a discrete one, representing it in digital form using a digital readout device (DDU) and can output information to an external device - display, digital printing. The advantages of digital measuring instruments (DIM) are:

Automatic selection of measuring range;

Automatic measurement process;

Outputting information in code to external devices;

Presentation of measurement results with high accuracy.

Great Russian scientist Dmitry Ivanovich Mendeleev said: “Science begins where measurements begin.” During this lesson, you will learn what a measurement is, what the scale division of a measuring instrument is and how to calculate it, and also learn how to determine the error (inaccuracy) of measurement results.

Topic: Introduction

Lesson No. 2: Physical quantities and their measurement.

Accuracy and error of measurements.

The purpose of the lesson: get acquainted with the concept of “physical quantities”; learn to measure physical quantities using simple measuring instruments and determine the measurement error.

Equipment: ruler, beaker, thermometer, ammeter, voltmeter.

1. Check homework(15 minutes).

1) The first student solves problem No. 5 at the board.

2) The second student solves problem No. 6 at the board.

3) The rest write a physical dictation.

4) How to ask additional questions problem solvers The board has questions for the paragraph and basic definitions.

6) As an additional question, ask 7 “A” about the messages on the piece of paper (what conclusions were drawn).

2. Studying new material (20 minutes).

You already know that to study various physical phenomena occurring with different physical bodies, you have to do experiments. And during experiments, it is necessary to measure various physical quantities, such as body mass, speed, time, height, length, width, etc. To measure physical quantities, various physical instruments are required.

2.1. What does it mean to measure a physical quantity?

(PZ): Measure a physical quantity - this means comparing it with another similar (as they say, homogeneous) physical quantity taken as a unit.

For example, the length of an object is compared to a unit of length, the mass of a body is compared to a unit of mass. But if one researcher measures the length, for example, of the distance traveled in fathoms, and another researcher measures it in feet, then it will probably be difficult for them to immediately understand each other.

Therefore, all over the world they try to measure physical quantities in the same units. In 1963, the International System of Units SI (SI - System International) was adopted. And it is in this system of units of measurement of physical quantities that we will continue to work.

For example, the most common physical quantities are length, mass and time. The International System of Units SI accepts:

Measure length in meters (m); unit of measurement – ​​1 m;

Measure mass in kilograms (kg), unit of measurement – ​​1 kg;

Time is measured in seconds (s), the unit of measurement is 1 s.

Of course, you know other, secondary units of measurement. For example, time can be measured in minutes or hours. But it is important to take into account that we will try to carry out all our subsequent calculations in the SI system.

Units that are 10, 100, 1000, 1,000,000, etc. times larger than the accepted units (so-called multiples) are often used.

For example: deca (dk) – 10, hecto (g) – 100, kilo (k) – 1000, mega (M) – 1,000,000, deci (d) – 0.1, centi (s) – 0.01, miles ( m) – 0.001.

Example: table length is 95 cm. Necessary V Express the length in meters (m)?

60 cm = 60 * 0.01 = 0.6 m

2.2. Measuring instrument scale division value

When taking measurements, it is very important to use measuring instruments correctly. You are already familiar with some instruments, such as a ruler and a thermometer. You have yet to get acquainted with others - the measuring cylinder, the voltmeter and the ammeter. But all these devices have one thing in common: they have a scale.

To work correctly with a measuring device, you must first pay attention to its measuring scale.

For example, consider the measuring scale of a very ordinary ruler.

Let's look at the ruler example in class together.

Using this ruler you can measure the length of any object, but not in SI units, but in centimeters. The scale of any device must indicate the units of measurement.

On the scale you see strokes (this is the name given to the lines marked on the scale). The spaces between the strokes are called scale divisions. Don't confuse strokes with divisions!

There are numbers next to some of the strokes.

In order to start working with any device, it is necessary to determine the scale division value of this device.

(PZ): The scale division value of a measuring instrument is the distance between the nearest scale strokes, expressed in units of the measured value. (in centimeters or millimeters for a ruler, in degrees for a thermometer, etc.).

To determine the value of a scale division of any measuring instrument, you need to select the two closest lines, next to which the numerical values ​​of the value are written. For example, two and one. Now you need to subtract the smaller from the larger value. The result must be divided by the number of divisions between the selected strokes

In our example, a student ruler.

Another example is a thermometer scale.

Rice. 2. Thermometer scale

We select the two nearest strokes with numbers, for example, 20 and 10 degrees Celsius (note that this scale also shows units of measurement, °C). There are 2 divisions between the selected strokes. Thus, we get

2.3. Measurement error and its determination.

To carry out measurements correctly, it is not enough to be able to determine the value of the instrument scale division. Remember that when talking about the distance from one point to another, we sometimes use expressions like “plus or minus half a kilometer.” This means that we do not know the exact distance, that in its measurement there was some inaccuracy, or, as they say, an error.

There is an error in any measurement; absolutely accurate instruments do not exist. And the magnitude of the error can also be determined by the scale of the measuring device.

(PZ): Measurement error is half the scale division of the measuring device.

Example 1. For example, a regular student ruler has a division value of 1 mm. Suppose we used it to measure the thickness of a piece of chalk and we got 12 mm. Half the price of a ruler division of 0.5 mm. This is the measurement error. If we denote the thickness of a piece of chalk by the letter b, then the measurement result is written as follows:

b = 12 + 0.5(mm)

The sign (plus or minus) means that during the measurement we could have made a mistake either up or down, that is, the width of a piece of chalk ranges from 11.5 mm to 12.5 mm.

I draw example No. 2 on the board with a smaller number of divisions, together with the class we calculate the central value and find the error.

Rice. 1. Regular ruler scale

CD = (2cm – 1cm)/5cm = 0.2cm = 2mm

Half the price of the ruler division in this case will be equal to 1 mm.

Then the width of the piece of chalk is b = 12 + 1(mm), that is, in this case, the width of a piece of chalk ranges from 11 mm to 13 mm. The scatter of measurements turned out to be larger.

In both cases, we made correct measurements, but in the first case the measurement error was smaller and the accuracy was higher than in the second, since the cost of dividing the ruler was less.

So from these two examples we can conclude:

(PZ): The lower the scale division of the device, the greater the accuracy (less error) of measurements using this device.

When recording values, taking into account the error, use the formula:

(PZ): A = a + ∆a,

where A is the measured quantity, a is the measurement result, ∆a is the measurement error.

3. Consolidation of the studied material (10 minutes).

Textbook: Exercise No. 1.

4. Homework.

Textbook: § 4, 5.

Problem book: No. 17, No. 39. ( detailed description tasks)

(explain how to write detailed solution tasks!!!)

The measured values ​​cannot be determined absolutely reliably. Measuring instruments and systems always have some tolerance and noise, which is expressed as a degree of inaccuracy. In addition, it is necessary to take into account the characteristics of specific devices.

The following terms are often used in relation to measurement uncertainty:

• Error- error between true and measured value
• Accuracy- random scatter of measured values ​​around their average
• Permission- the smallest distinguishable value of the measured value

Often these terms are confused. Therefore, here I would like to discuss the above concepts in detail.

Measurement uncertainty

Measurement inaccuracies can be divided into systematic and random measurement errors. Systematic errors are caused by deviations in gain and zero adjustment of the measuring equipment. Random errors are caused by noise and/or currents.

Often the concepts of error and accuracy are considered synonymous. However, these terms have completely different meanings. The error shows how close the measured value is to its real value, that is, the deviation between the measured and actual value. Accuracy refers to the random variation of measured quantities.

When we carry out a certain number of measurements until the voltage or some other parameter stabilizes, then some variation will be observed in the measured values. This is caused by thermal noise in the measuring circuit of the measuring equipment and measuring setup. Below, the left graph shows these changes.

Definitions of uncertainties. On the left is a series of measurements. On the right are the values ​​in the form of a histogram.

bar chart

The measured values ​​can be plotted as a histogram, as shown on the right in the figure. The histogram shows how often a measured value is observed. The highest point on the histogram, this is the most frequently observed measured value, and in the case of a symmetric distribution is equal to the mean value (depicted by the blue line in both graphs). The black line represents the true value of the parameter. The difference between the average of the measured value and the true value is the error. The width of the histogram shows the spread of individual measurements. This spread of measurements is called precision.

Use the correct terms

Accuracy and accuracy therefore have different meanings. Therefore, it is possible that the measurement is very accurate, but has an error. Or vice versa, with a small error, but not accurate. In general, a measurement is considered reliable if it is accurate and has little error.

Error

The error is an indicator of the correctness of the measurement. Due to the fact that in one measurement the accuracy affects the error, the average of a series of measurements is taken into account.

The accuracy of a measuring instrument is usually specified in two values: the error of indication and the full-scale error. These two characteristics together determine the overall measurement error. These measurement error values ​​are expressed as a percentage or ppm (parts per million, parts per million) relative to the current national standard. 1% corresponds to 10000 ppm.

Accuracy is given for specified temperature ranges and for a specified period of time after calibration. Please note that in different ranges, different errors are possible.

Indication error

The indication of percentage deviation without further specification also applies to the indication. Voltage divider tolerances, amplification precision, and readout and digitization absolute tolerances are the causes of this error.

5% inaccuracy for 70V

A voltmeter that reads 70.00V and has a specification of "±5% of reading" will have an error of ±3.5V (5% of 70V). The actual voltage will be between 66.5 and 73.5 volts.

Full scale error

This type of error is caused by offset errors and linearity errors of the amplifiers. For devices that digitize signals, there is nonlinearity of conversion and ADC errors. This characteristic applies to the entire usable measuring range.

The voltmeter may have a “3% scale” characteristic. If the 100 V range (equal to full scale) is selected during measurement, the error is 3% of 100 V = 3 V, regardless of the measured voltage. If the reading in this range is 70 V, then the actual voltage lies between 67 and 73 volts.

3% span error in 100 V range

It is clear from the above figure that this type of tolerance is independent of the readings. When reading 0 V, the actual voltage lies between -3 and 3 volts.

Scale error in numbers

Often for digital multimeters the scale error is given in digits instead of as a percentage.

For a digital multimeter with a 3½-digit display (range -1999 to 1999), the specification may indicate “+ 2 digits.” This means that the reading error is 2 units. For example: if the range is 20 volts (±19.99), then the scale error is ±0.02 V. The display shows a value of 10.00, but the actual value will be between 9.98 and 10.02 volts.

Calculation of measurement error

The indication and scale tolerance specifications together determine the overall measurement uncertainty of the instrument. The calculation below uses the same values ​​as in the examples above:

Accuracy: ±5% reading (3% span)

Range: 100V

Reading: 70 V

The total measurement error is calculated as follows:

In this case, the total error is ±6.5V. The true value lies between 63.5 and 76.5 volts. The figure below shows this graphically.

Total inaccuracy for 5% and 3% span reading inaccuracies for 100 V range and 70 V reading

Percentage error is the ratio of error to reading. For our case:

Numbers

Digital multimeters may have a specification of "±2.0% reading, +4 digits". This means that 4 digits must be added to the 2% reading error. As an example, consider again a 3½-digit digital indicator. It reads 5.00 V for the selected 20 V range. 2% of the reading would mean an error of 0.1 V. Add to this the numerical error (= 0.04 V). The total error is therefore 0.14 V. The true value should be between 4.86 and 5.14 volts.

Total error

Often, only the error of the measuring device is taken into account. But also, the errors of measuring instruments, if they are used, should additionally be taken into account. Here are some examples:

Increased error when using a 1:10 probe

If a 1:10 probe is used in the measurement process, then it is necessary to take into account not only the measuring error of the device. The accuracy is also affected by the input impedance of the device used and the resistance of the probe, which together make up the voltage divider.

The figure above shows a schematic with a 1:1 probe connected to it. If we consider this probe as ideal (no connection resistance), then the applied voltage is transferred directly to the input of the oscilloscope. The measurement error is now determined only by the permissible deviations of the attenuator, amplifier and circuits taking part in further signal processing and is set by the device manufacturer. (The error is also affected by the connection resistance that forms internal resistance. It is included in the specified permissible deviations).

The picture below shows the same oscilloscope, but now a 1:10 probe is connected to the input. This probe has an internal connection resistance and, together with the oscilloscope's input resistance, forms a voltage divider. The permissible deviation of the resistors in the voltage divider is the cause of its own error.

A 1:10 probe connected to an oscilloscope introduces additional uncertainty

The oscilloscope's input impedance tolerance can be found in its specification. The permissible deviation of the probe connection resistance is not always given. However, system accuracy is stated by the manufacturer of a specific oscilloscope probe for a specific type of oscilloscope. If the probe is used with a different type of oscilloscope than the recommended one, then the measurement error becomes uncertain. You should always try to avoid this.

Let's assume that the oscilloscope has a tolerance of 1.5% and is using a 1:10 probe with a system error of 2.5%. These two characteristics can be multiplied to obtain the total error of the instrument reading:

Here is the total error of the measuring system, - the error of the instrument reading, - the error of a probe connected to an oscilloscope of a suitable type.

Measurements with a shunt resistor

An external shunt resistor is often used when measuring currents. The shunt has some tolerance that affects the measurement.

The specified tolerance of the shunt resistor affects the reading error. To find the total error, the permissible deviation of the shunt and the error of the measuring device are multiplied:

In this example, the total reading error is 3.53%.

The shunt resistance depends on temperature. The resistance value is determined for a given temperature. Temperature dependence is often expressed in .

For example, let's calculate the resistance value for temperature environment. The shunt has the following characteristics: Ohm(respectively and ) and temperature dependence .

The current flowing through the shunt causes energy to dissipate on the shunt, which leads to an increase in temperature and, consequently, to a change in the resistance value. The change in resistance value when current flows depends on several factors. To make a very accurate measurement, it is necessary to calibrate the shunt for resistance drift and the environmental conditions under which the measurements are taken.

Accuracy

Term accuracy used to express the randomness of measurement error. The random nature of deviations of measured values ​​in most cases is of a thermal nature. Due to the random nature of this noise, it is not possible to obtain an absolute error. Accuracy is given only by the probability that the measured quantity lies within certain limits.

Gaussian distribution

Thermal noise has a Gaussian, or, as they also say, normal distribution . It is described by the following expression:

Here is the average value, shows the dispersion and corresponds to the noise signal. The function gives a probability distribution curve as shown in the figure below, where the mean and effective noise amplitude are .

And

The table shows the chances of obtaining values ​​within the specified limits.

As you can see, the probability that the measured value lies in the range ± is equal to .

Increased accuracy

Accuracy can be improved by oversampling (changing the sampling rate) or filtering. Individual measurements are averaged, so noise is greatly reduced. The spread of measured values ​​is also reduced. When using resampling or filtering, it must be taken into account that this may lead to a decrease in throughput.

Permission

Permission, or, as they also say, resolution of a measuring system is the smallest discernible measurand. Determining the resolution of an instrument does not refer to the accuracy of the measurement.

Digital measuring systems

A digital system converts an analog signal into a digital equivalent using an analog-to-digital converter. The difference between the two values, that is, the resolution, is always one bit. Or, in the case of a digital multimeter, it's one digit.

It is also possible to express resolution in terms of units other than bits. As an example, consider having an 8-bit ADC. The vertical sensitivity is set to 100 mV/div and the number of divisions is 8, the total range is therefore 800 mV. 8 bits are represented 2 8 =256 different meanings. The resolution in volts is then equal to 800 mV / 256 = 3125 mV.

Analog measurement systems

In the case of an analog instrument, where the measured value is displayed mechanically, as in a pointer instrument, it is difficult to obtain an exact number for resolution. First, resolution is limited by mechanical hysteresis caused by friction in the pointer mechanism. On the other hand, resolution is determined by the observer making his subjective assessment.