INC 253 Digital and electronics laboratory Iinc.kmutt.ac.th/course/inc253/Lab01-Basic Errors in Measurements.pdf · INC 253 Digital and electronics laboratory I Laboratory I ... Precautions:

Department of Control System and Instrumentation Engineering

Faculty of Engineering

King Mongkut’s University of Technology Thonburi

INC 253 Digital and electronics laboratory I

Laboratory I

Basic Errors in Measurements

Author: …………………………… ID …………………

Co-Authors:

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2. …………………………… ID …………………

3. …………………………… ID …………………

Experiment Date: ………………… Report received Date: …………………

Comments

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For Instructor

Full Marks

Pre lab 10

Results 15

Discussion 25

Questions 10

Conclusion 5

Total 65

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INC, KMUTT Lab. 1: Basic Errors in Measurements

Objectives

1. To study the use of measuring instrument to measure resistance, voltage and current.

2. To study the non-ideal characteristics of electrical measurement equipment.

3. To become familiar with concepts relating to error analysis and uncertainty.

Equipments Required

1. Digital multimeter (Advantest R6451A)

2. Analog multimeter

3. DC Power Supply

4. Circuit construction breadboard

Devices Required

1. Resistors: 120 kΩ , 5 % tolerance, (1)

2. Resistors: 470 kΩ , 5 % tolerance, (1)

3. Resistors: 1.5 kΩ , 5 % tolerance, (1)

Basic Information

1. DC voltage measurements

A voltage exits as a potential difference across two points of an energized circuit. To correctly

measure a voltage, a voltmeter must be connected across the two points of interest. The positive

or red test lead is connected to whichever point is closest in potential to the positive side of the

energy source. The negative or black test lead connects to the other point.

As shown in Figure 1, a single subscripted voltage, such as Va, refers to the potential from

point “a” to a ground or circuit common reference point. Double subscripted voltages, such as

Vbc, refers to the potential from point “b” to point “c”. The positive lead must be connected to

point “b”, the negative to point “c”. An up scale reading is a positive voltage, if deflection is

downwards, the meter leads must be reversed, and the resulting measurement must be recorded

as a negative voltage.

Precaution: When measuring voltages, start on a high voltage range and successively reduce the

range to obtain as large a deflection as possible without causing the needle to exceed maximum

deflection.

3


R1

R2 R3V

+ - COMVOLTMETER

a

VaVbc

c

b

Figure 1 Reference voltage

2. DC current measurements

The flow of current in a circuit is much like the flow of water through a pipe. For this reason,

the current must be made to pass through the meter to make a measurement. Referring to

Figure 2, the measured circuit must be physically broken before a current measurement can be

accomplished. When a circuit is broken at a location, it leaves a hole with two wires ends

available. The ammeter must be connected to each of these wire ends, thus completing the

circuit again. The positive lead of the ammeter should be connected to the wire end nearest to

the positive side of the energy source.

Precautions:

a) Be sure to break the circuit and insert the ammeter in resulting hole. Start on the high

range and successively reduce ranges until a reasonably high deflection is obtained.

b) Be extremely careful to connect the ammeter in series with the line in which a

measurement is required; never connect the ammeter in parallel with a component.

R1

R2V

AMMETER

I

+ - COM

Figure 2 using an ammeter to measure circuit current.

3. Resistance measurements

Ohmmeters are used to measure the resistance of a component or a group of components.

Ohmmeters differ significantly from voltmeters or ammeters in that they used their own power

supply to accomplish a resistance measurement. Both voltmeters and ammeters used power

from the circuit under test, thus the circuit must remain energized. A circuit must be de-

energized before a resistance measurement can be made. Simply opening the circuit where it is

desired to measure resistance can do this. Care must be taken that the circuit is not closed by

the ohmmeter itself.

4


R1

R2R3V

- COM

OHMMETER+

Figure 3 an ohmmeter measures net resistance between meter terminals.

4. Errors

Every measurement that is made is subject to a number of errors. The following is a list of

possible sources of error:

4.1. Insertion and Loading Errors

An important rule in making any measurement is that the measuring process must not

significantly upset and alter the phenomena being measured. In practice, the measuring process

will have some effect on the measurement being made, and this is something that should be

considered as a source of error. Such errors may be caused by “inserting” an ammeter with

non-zero impedance in a circuit, or placing a voltmeter across a circuit that “loads” down the

voltage being measured.

4.2. Instrument Error

Any measuring instrument will be accurate only to a certain extent and only if it has been

calibrated. This is true for the measuring instruments used in this course. For example, suppose

that a Simpson 260 model and a Fluke 8010A model multimeter are used separately to measure

a dc voltage of 1.78 V. If the 2.5 VDC range on the Simpson is used, there will be an inherent

error of ± 2 % of full scale deflection or ± 0.05 V (2.5 V x 2%). The result will be (1.78 ± 0.05)

V. If the 2 VDC range of the Fluke is used, the error introduced will be ± (0.1% of the reading

+ one digit. If the Fluke’s digital display shows 1.782 V, then the error due to the actual reading

is 0.001782 V. The readout, however, cannot display all these digits and the error is rounded

off to 0.002. An additional error is introduced in the last or least significant digit (LSD) of 1(=

0.001 V for 2 V range, there are three digits after the decimal point in the meter’s display),

giving a total error in the reading of 0.003 V. The recorded reading now becomes (1.782 ±

0.003) V.

4.3. Human Error

A human error is an error made by an observer when recording a measurement or by using an

instrument incorrectly. Two types of human error are parallax reading error (caused by reading

an instrument pointer from an angle) and interpolation error (made in “guessing” the correct

value between two calibrated marks on the meter scale).

5. Significant Figures

It is customary in measurement work to report a result with all the digits of which we are sure,

and a final digit that is believed to be nearest to the true value. For example, a measurement of

"5.17 V" indicates that the voltage is known to better than one tenth of a volt and that the

measurement of the hundredth of a volt is uncertain but is close to seven. Superfluous figures

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are sometimes allowed to accumulate in computations involving addition, subtraction,

multiplication or division, but the final answer must be rounded off if extra digits accumulate

beyond the inherent limits of certainty of the original data. It is standard practice that if the digit

to be discarded is less than 5, then it and any succeeding numbers are dropped; if the digit is 5

or greater, then the previous digit is increased by one and succeeding numbers are dropped. For

example, if 67.238 were to be rounded to three significant digits, then it would rounded to 67.2;

if 67.267 were to be rounded to three significant digits, then it would rounded to 67.3. Some

examples of accumulation of superfluous figures and rounding off are given below.

Example 1.1

Suppose we have two resistances in series with values of R1 = 24.4 Ω and R2 = 0.516 Ω. The

total series resistance (R1 + R2) is 24.916 Ω, but this value is meaningless since R1 has an

accuracy of ± 0.1 Ω and the sum cannot be guaranteed to a one-tenth of an ohm. The sum

should therefore be rounded to 24.9 Ω.

Example 1.2

Suppose we wish to find the voltage drop across a resistor using Ohm's law and the values

measured for resistance and current are 45.73 Ω and 2.56 A respectively. The voltage drop

becomes V = IR = 117.0688 V. The resistance value is known to four significant figures and

the current is known to three. This limits the number of significant figures that can be included

in the answer since the answer cannot be known to greater accuracy than the least defined of

the parameters. The correct procedure is therefore to round off the answer to three significant

figures, i.e. to 117 V.

6. Limiting Errors (Guaranteed Accuracy)

Suppose the power dissipated in a 100 Ω, 5 % resistor is to be determined using P = I2R where

I = 4.6 A as measured on the 10 A range of an ammeter with a full scale deflection (fsd)

accuracy of ± 2 %. The manufacturer's specification of the tolerance on the value of resistance

does not specify a standard deviation, but instead specifies that the error will be no greater than

5 % (5 Ω in our example). Similarly, the manufacturer of the ammeter guarantees that the error

in the current measurement will be no greater than ± 2 % fsd (0.2 A). These are limiting errors

in the sense that it is virtually certain that the errors will not exceed these limits.

7. Propagation of Uncertainty

Suppose that in addition to calculating the power P we wish to calculate the limits within which

the value of P can be guaranteed. The maximum value of P can be calculated using methods of

partial differentiation by treating the uncertainty in each variable as a differential and the

uncertainty in the derived result as a total differential.

The total differential for a function w = f(x, y) is defined as

yy

wx

x

ww ddd

(1.1)

where dx and dy are differentials in x and y respectively. If the uncertainties in P, I, and R can

be expressed as p = dw, i = dx, and r = dy, then the expression for the limit of

uncertainty in P can be written as

rr

pi

i

pp

(1.2)

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where absolute values are used to allow for the worst case where the effects of the uncertainties

are additive.

Example 1.3 Find power dissipation P in resistor R when the direct current I is passing through it. When the

measured value of resistance is R = 𝑟 ± ∆𝑟 = 100 ± 5 Ω and of current is 𝐼 = 𝑖 ± ∆𝑖 = 4.6 ± 0.2 𝐴. Then report the result.

𝑃 = 𝑝 ± ∆𝑝 = 𝐼2𝑅

Since

𝑝 = 𝑖2𝑟 = 4.6 2 100 = 2 116 W (1.3)

the partial derivatives are

iri

ri

i

p2

)( 2

2

2 )(i

r

ri

r

p

(1.4)

If i = 0.2 A and r = 5 Ω, then the uncertainty in P can be evaluated using equation (1.2)

W8.2898.105184)0.5()6.4()2.0)(1006.42( 2 xxp (1.5)

We will report the power dissipation by

Round off the uncertainty in equation (1.5) to 1 or 2 significant figures.

Round off the answer in equation (1.3) so it has the same number of digits before or after the

decimal point as the answer.

Put the answer and its uncertainty in parentheses,

Then put the power of 10 and unit outside the parentheses.

𝑃 = 2116 ± 289.8 = 21.2 ± 2.9 × 102 W

Then the maximum percentage error is 2.9 21.2 = 0.1368 = 13.6 %. It should be noted

that the actual error will be very likely less than the maximum error (worst case).

The propagated uncertainty for P can also be calculated using relative errors. If

rIiirrr

pi

i

pp

22 (1.6)

then an expression for the relative uncertainty in power can be obtained by dividing both sides

of the equation by p = i2r:

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r

r

i

i

p

p

2 (1.7)

where p/p, i/i, and r/r are the relative uncertainties in P, I, and R respectively. Again, the

maximum percentage error is 13.6 %.

Pre-Lab Preparation

The lab preparation must be completed before doing the experiment. Show it to your instructor

at the beginning of the lab.

1. Resistor uncertainties

a) Given the resistors having coded color as specified in Table 1 on the Lab Measurements

Sheet. Fill in the space with the corresponding values referring the 1st row values as an example.

b) You will use three resistors in the lab having nominal values of 1.5 kΩ, 120 kΩ and 470

kΩ. If each resistor has a tolerance of ± 5 %, calculate the uncertainties (Maximum permissible

error) in the resistors. Enter the calculated values in Table 2.

2. Voltmeters

a) Figure 4 shows an electrical circuit consisting of a DC voltage source V1, two across R2.

If V1 = 10.0 V, R1 = 470(1 ± 5 %) kΩ and R2 = 120(1 ± 5 %) kΩ and voltmeter VM is an ideal

voltmeter. Calculate the expected reading of the meter (VM

), and record it in the column labeled

“Calculated value” in Table 6 on the Lab Measurements Sheet.

Calculate the uncertainty in VM

due to the uncertainty of the given resistors. Show all of your

calculations on a separate sheet of paper.

b) Calculate the expected reading (VDIGITAL

) when a practical digital multimeter (Advantest

R6451A) set to measure dc voltage is used instead of an ideal meter for VM. Record your

results in Table 6. Note that the multimeter, when used as a voltmeter, can be modeled as an

ideal voltmeter in parallel with an internal input resistance (Rin) as shown in Figure 5. The

value of the resistance is dependent on the scale setting used. Include the internal input

resistance of the meter in your calculations. The values needed for this calculation can be found

from document provided.

Figure 4 Voltage measurement circuit Figure 5 Using non-ideal meter

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c) Repeat step b) using a practical analog meter (Your meter, ________ Model_________)

instead of a practical digital multimeter, i.e., calculate VANALOG., assuming a full scale

deflection of 2.5 V DC, the sensitivity is 20,000 Ω/V. Hence, the input resistance is 2.5 V x

20,000 Ω/V = 50,000 Ω = 50 kΩ. Record VANALOG in Table 6.

3. Ammeters

a) Figure 6 shows an electrical circuit consisting of a DC voltage source V1, a resistor R and

an ammeter AM connected in series. If R = 1.5(1 ± 5 %) kΩ and ammeter AM is an ideal

ammeter, then calculate the expected reading of the meter for a DC source voltage V1 = 1.3 V.

Record your results in the column labeled “Calculated value” in Table 9 of the Lab

Measurements Sheet.

Also calculate the uncertainty in the expected reading due to the uncertainty of the given

resistor. Show your calculations on a separate sheet of paper.

Figure 6 Current measurement circuit

b) Re-calculate the current if a practical digital multimeter (Advantest) is used instead of an

ideal meter as shown in Figure 7. Include the internal input resistance of the meter in your

calculations. The values needed for this calculation can be found from document provided.

Record your results in the column labeled “Calculated value” (ADIGITAL

) in Table 9 of the Lab

Measurements Sheet.

9


Figure 7 Current measurements using a non-ideal meter

The internal resistance of meters in the ammeter mode can be calculated using the voltage

drops at full scale deflection obtained from the instrument specifications. For example, the

specification for the Simpson Model 260 meter shows a voltage drop of 280 mV fsd when the

meter is set for the 0 mA to 100 mA current range. The ammeter resistance for this range is

therefore:

8.2A10100

V102803

3

Rin (3.8)

c) Repeat step b) but use an analog multimeter (Each individual multi meter) as

an ammeter in the appropriate dc current measurement setting instead of the digital multimeter,

i.e., calculate AANALOG.Record AANALOG in Table 9.

Procedure

Note the following information before starting the lab:

a) You will be using two different meters for this lab exercise: a digital multimeter and your

own analog multimeter.

b) When using either the digital and/or analog meter to do any measurement, set it to the

range that will allow you to obtain the maximum digits or deflection.

1. Ohmmeters

You are provided with three resistors to be used in this and other parts of the lab exercise.

These have nominal values of 1.5 kΩ, 120 kΩ, and 470 kΩ.

Complete data from specification of each meter in table 3

Measure the value of each resistor using both the digital and your own analog multimeter and

then report the results (To obtain individual data) in Table 4 on the Lab Measurements Sheet.

2. Voltmeters

a) Connect the circuit shown in Figure 4 using the breadboard. Set dc voltage V1 = 10.0 V

using the digital multimeter (assuming that the voltage is exact).

b) Complete data from specification of each voltmeter in table 5. Also, calculate and record

in Table 5 the corresponding meter uncertainty.

c) Measure VM using the digital multimeter set to the range that will allow you to obtain the

maximum digits after the decimal point. Record the readings in Table 6(VDIGITAL

).

d) Repeat part b) using the analog multimeter set to the range that will allow you to obtain

the maximum scale deflection. Record the reading in Table 6 (VANALOG

).

e) Report the results in table 7 on the Lab Measurements Sheet.

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3. Ammeters

a) Connect the circuit shown in Figure 6 using the circuit construction breadboard. Set dc

voltage V1 = 1.3 V using the digital multimeter (assuming that the voltage is exact).

b) Complete data from specification of each voltmeter in table 8. Also calculate the

corresponding ammeter uncertainty.

c) Measure the current using the digital multimeter set to the range that will allow you to

obtain the maximum digits after the decimal point. Record the readings in Table 9 (ADIGITAL

).

d) Repeat step c) using individual student multimeter set to the range that will allow you to

obtain the maximum scale deflection. Record the readings in Table 9 (AANALOG

).

e) Report the results in table 10 on the Lab Measurements Sheet.

11


Lab Measurements Sheet

Sections marked * is pre-lab preparation and must be completed BEFORE doing experiment.

Table 1 Color-coded Resistors and resistor uncertainty

First

band

Second

band

Third

band

Fourth

band

Resistor value Resistance range,

Ω Ω %

Tolerance

Orange Orange Brown No color 330 20 264 - 396

Gray Red Gold Silver

Yellow Violet Green Gold

Orange White Orange Gold

Green Blue Brown No color

Red Red Yellow Silver

Brown Green Gold Gold

Blue Gray Green No color

Green Black Silver Gold

Part A Resistance measurement

Table 2 Possible resistance values

Resistor Nominal value (k) * Uncertainty (k)

R1 1.5

R2 120

R3 470

Table 3 About the meters

Digital ohmmeter Analog ohmmeter

Manufacturer

Range

Accuracy specification

Meter uncertainty

12


Table 4 Report the results

Resistor Result of measurement

Digital ohmmeter Analog ohmmeter

R1

R2

R3

Result of measurement = (measured value ± meter uncertainty)

13


Part B Voltage Measurement

Table 5 about the voltmeters

Digital voltmeter Analog voltmeter

Manufacturer

Range


Meter uncertainty

Input resistance

Table 6 Voltage measurements

Voltage

measurement

*Calculated

value (V)

Measured value

(V)

% Error from

ideal

% Error from

calculated

VM (ideal) --------------- --------------- --------------

VDIGITAL

VANALOG

Table 7 Report the result

Result of measurement

Digital voltmeter Analog voltmeter

“% Error from ideal” is calculated using the formula:

100%

valueideal

valueidealvaluemeasuredError

“% Error from calculated” is calculated using the formula:

100%

valuecalculated

valuecalculatedvaluemeasuredError

14


Part C Current Measurement

Table 8 About the ammeters

Table 9 Current measurements

Current

measurement

*Calculated

value (mA)

Measured

value (mA)

% Error from ideal % Error from

calculated

AM (ideal) ------------- ---------------- ----------------

ADIGITAL

AANALOG

Table 10 Report the results

Result of measurement

Digital ammeter Analog ammeter

Questions

1. In part A, are the measured values of resistance within the tolerances specified by

manufacturer?

2. In part B, when measuring the voltage across a resister, should the input resistance of the

meter be much lower or higher than the resistance of the resistor? Why?

3. In part C, when measuring the current through a resister, should the input resistance of the

meter be much lower or higher than the resistance of the resistor? Why?

Digital ammeter Analog ammeter

Range


Meter uncertainty

Meter voltage drop

Documents

INC 253 Digital and electronics laboratory Iinc.kmutt.ac.th/course/inc253/Lab01-Basic Errors in Measurements.pdf · INC 253 Digital and electronics laboratory I Laboratory I ... Precautions: