inc 253 digital and electronics laboratory iinc.kmutt.ac.th/course/inc253/lab01-basic errors in...
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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:
1. …………………………… ID …………………
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.
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INC, KMUTT Lab. 1: Basic Errors in Measurements
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.
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INC, KMUTT Lab. 1: Basic Errors in Measurements
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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INC, KMUTT Lab. 1: Basic Errors in Measurements
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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INC, KMUTT Lab. 1: Basic Errors in Measurements
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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INC, KMUTT Lab. 1: Basic Errors in Measurements
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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INC, KMUTT Lab. 1: Basic Errors in Measurements
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.
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INC, KMUTT Lab. 1: Basic Errors in Measurements
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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INC, KMUTT Lab. 1: Basic Errors in Measurements
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.
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INC, KMUTT Lab. 1: Basic Errors in Measurements
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
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INC, KMUTT Lab. 1: Basic Errors in Measurements
Table 4 Report the results
Resistor Result of measurement
Digital ohmmeter Analog ohmmeter
R1
R2
R3
Result of measurement = (measured value ± meter uncertainty)
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INC, KMUTT Lab. 1: Basic Errors in Measurements
Part B Voltage Measurement
Table 5 about the voltmeters
Digital voltmeter Analog voltmeter
Manufacturer
Range
Accuracy specification
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
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INC, KMUTT Lab. 1: Basic Errors in Measurements
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
Accuracy specification
Meter uncertainty
Meter voltage drop