Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition

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6 Measurement and Data Analysis for Engineering and Science

along the length, L, of a pipe of radius, R, for a liquid with absolute viscosity,

µ, ﬂowing at a volumetric ﬂow rate, Q. In this experiment, the volumetric

ﬂow rate is varied. Thus, µ, R, and L are parameters, Q is the independent

variable, and ∆p is the dependent variable. R, L, Q, and ∆p are measur-

ands. The viscosity is a dependent variable that is determined from the ﬂuid’s

temperature (another measurand and parameter). If the ﬂuid’s temperature

is not controlled in this experiment, it could aﬀect the values of density and

viscosity, and hence aﬀect the values of the dependent variables.

Example Problem 1.1

Statement: An experiment is performed to determine the coeﬃcient of restitution, e,

of a ball over a range of impact velocities. For impact normal to the surface, e = v

where v

is the normal velocity component immediately before impact with the surface

and v

is that immediately after impact. The velocity, v

, is controlled by dropping the

ball from a known initial height, h

, and then measuring its return height, h

. What

are the variables in this experiment? List which ones are independent, dependent,

parameter, and measurand.

Solution: A ball dropped from height h

will have v

√

2gh

, where g is the local

gravitational acceleration. Because v

√

2gh

, e =

. So the variables are

, h

, v

, e, and g. h

is an independent variable; h

, v

, and e are dependent

variables; h

and g are parameters; h

and h

are measurands.

Often, however, it is diﬃcult to identify and control all of the variables

that can inﬂuence an experimental result. Experiments involving biologi-

cal systems often fall into this category. In these situations, repeated mea-

surements are performed to arrive at statistical estimates of the measured

variables, such as their means and standard deviations. Repetition implies

that a set of measurements are repeated under the same, ﬁxed operating

conditions. This yields direct quantiﬁcation of the variations that occur in

the measured variables for the same experiment under ﬁxed operating con-

ditions. Often, however, the same experiment may be run under the same

operating conditions at diﬀerent times or places using the same or compa-

rable equipment and facilities. Because uncontrollable changes may occur in

the interim between running the experiments, additional variations in the

measured variables may be introduced. These variations can be quantiﬁed

by the replication (duplication) of the experiment. A control experiment

is an experiment that is as nearly identical to the subject experiment as

possible. Control experiments typically are performed to reconﬁrm a sub-

ject experiment’s results or to verify a new experimental set-up’s perfor-

mance. Finally, experiments can be categorized broadly into timewise and

sample-to-sample experiments [10]. Values of a measurand are recorded

in a continuous manner over a period of time in timewise experiments. Val-

ues are obtained for multiple samples of a measurand in sample-to-sample

experiments. Both types of experiments can be considered the same when

values of a measurand are acquired at discrete times. Here, what distin-

guishes between the two categories is the time interval between samples.

Experiments 7

In the end, performing a good experiment involves identifying and con-

trolling as many variables as possible and making accurate and precise mea-

surements. The experiment always should be performed with an eye out for

discovery. To quote Sir Peter Medawar [6], “The merit of an experiment lies

principally in its design and in the critical spirit in which it is carried out.”

1.4 Experimental Approach

Park [7] remarks that “science is the systematic enterprise of gathering

knowledge about the world and organizing and condensing that knowledge

into testable laws and theories.” Experiments play a pivotal role in this

process. The general purpose of any experiment is to gain a better under-

standing about the process under investigation and, ultimately, to advance

science. Many issues need to be addressed in the phases preceding, during,

and following an experiment. These can be categorized as planning, design,

construction, debugging, execution, data analysis, and reporting of results

[10].

Prior to performing the experiment, a clear approach must be developed.

The objective of the experiment must be deﬁned along with its relation to

the theory of the process. What are the assumptions made in the experi-

ment? What are those made in the theory? Special attention should be given

to assuring that the experiment correctly reﬂects the theory. The process

should be observed with minimal intervention, keeping in mind that the

experiment itself may aﬀect the process. All of the variables involved in the

process should be identiﬁed. Which can be varied? Which can be controlled?

Which will be recorded and how? Next, what results are expected? Does the

experimental set-up perform as anticipated? Then, after all of this has been

considered, the experiment is performed.

Following the experiment, the results should be reviewed. Is there agree-

ment between the experimental results and the theory? If the answer is

yes, the results should be reconﬁrmed. If the answer is no, both the experi-

ment and the theory should be examined carefully. Any measured diﬀerences

should be explained in light of the uncertainties that are present in the ex-

periment and in the theory.

Finally, the new results should be summarized. They should be pre-

sented within the context of uncertainty and the limitations of the theory

and experiment. All this information should be presented such that another

investigator can follow what was described and repeat what was done.

8 Measurement and Data Analysis for Engineering and Science

1.5 Classiﬁcation of Experiments

There are many ways to classify experiments. One way is according to the

intent or purpose of the experiment. Following this approach, most exper-

iments can be classiﬁed as variational, validational, pedagogical, or

explorational.

The goal of variational experiments is to establish (quantify) the mathe-

matical relationships of the experiment’s variables. This is accomplished by

varying one or more of the variables and recording the results. Ideal vari-

ational experiments are those in which all the variables are identiﬁed and

controlled. Imperfect variational experiments are those in which some of the

variables are either identiﬁed or controlled. Experiments involving the de-

termination of material properties, component behavior, or system behavior

are variational. Standard testing also is variational.

Validational experiments are conducted to validate a speciﬁc hypothesis.

They serve to evaluate or improve existing theoretical models. A critical

validational experiment, which also is known as a Galilean experiment, is

designed to refute a null hypothesis. An example would be an experiment

designed to show that pressure does not remain constant when an ideal gas

under constant volume is subjected to an increase in temperature.

Pedagogical experiments are designed to teach the novice or to demon-

strate something that is already known. These are also known as Aris-

totelian experiments. Many experiments performed in primary and sec-

ondary schools are this type, such as the classic physics lab exercise designed

to determine the local gravitational constant by measuring the time it takes

a ball to fall a certain distance.

Explorational experiments are conducted to explore an idea or possible

theory. These usually are based upon some initial observations or a simple

theory. All of the variables may not be identiﬁed or controlled. The exper-

imenter usually is looking for trends in the data in hope of developing a

relationship between the variables. Richard Feynman [8] aptly summarizes

the role of experiments in developing a new theory, “In general we look for

a new law by the following process. First we guess it. Then we compute the

consequences of the guess to see what would be implied if this law that we

guessed is right. Then we compare the result of the computation to nature,

with experiment or experience, compare it directly with observation, to see

if it works. If it disagrees with experiment it is wrong. In that simple state-

ment is the key to science. It does not make any diﬀerence how beautiful

your guess is. It does not make any diﬀerence how smart you are, who made

the guess, or what his name is − if it disagrees with experiment it is wrong.

That is all there is to it.”

An additional ﬁfth category involves experiments that are far less com-

mon and lead to discovery. Discovery can be either anticipated by theory

Experiments 9

(an analytic discovery), such as the discovery of the quark, or serendipi-

tous (a synthetic discovery), such as the discovery of bacterial repression by

penicillin. There also are thought (gedunken or Kantian) experiments that

are posed to examine what would follow from a conjecture. Thought experi-

ments, according to our formal deﬁnition, are not experiments because they

do not involve any physical change in the process.

10 Measurement and Data Analysis for Engineering and Science

1.6 Problem Topic Summary

Topic

Review Problems

Homework Problems

Experiments

2, 3, 5, 6

1, 2, 3, 4, 5, 6, 7, 8

Variables

1, 4, 7

2, 4, 6, 9

TABLE 1.1

Chapter 1 Problem Summary

1.7 Review Problems

1. Variables manipulated by an experimenter are (a) independent, (b) de-

pendent, (c) extraneous, (d) parameters, or (e) presumed.

2. Immediately following the announcement by the University of Utah, on

March 23, 1989, that Stanley Pons and Martin Fleischmann had “dis-

covered” cold fusion, scientists throughout the world rushed to perform

an experiment that typically would be classiﬁed as (a) variational, (b)

validational, (c) pedagogical, (d) explorational, or (e) serendipitous.

3. If you were trying to perform a validational experiment to determine

the base unit of mass, the gram, which of the following ﬂuid conditions

would be most desirable? (a) a beaker of ice water, (b) a pot of boiling

water, (c) a graduated cylinder of water at room temperature, (d) a

thermometer ﬁlled with mercury.

4. Match the following with the most appropriate type of variable (indepen-

dent, dependent, extraneous, parameter, or measurand): (a) measured

during the experiment, (b) ﬁxed throughout the experiment, (c) not

controlled during the experiment, (d) aﬀected by a change made by the

experimenter, (e) changed by the experimenter.

5. What is the main purpose of the scientiﬁc method?

6. Classify the following experiments: (a) estimation of the heating value of

gasoline, (b) measuring the stress-strain relation of a new bio-material,

Experiments 11

7. An experiment is performed to determine the velocity proﬁle along a

wind tunnel’s test section using a pitot-static tube. The tunnel ﬂow rate

is ﬁxed during the experiment. Identify the independent, dependent, ex-

traneous, and parameter variables from the following list: (a) tunnel fan

revolutions per minute, (b) station position, (c) environment pressure

and temperature, (d) air density, (e) change in pressure measured by

the pitot-static tube, (f) calculated velocity.

1.8 Homework Problems

1. Give one historical example of an inductivistic, a fallibilistic, and a con-

ventionalistic experiment. State each of their signiﬁcant ﬁndings.

2. Write a brief description of an experiment that you have performed or

one with which you are familiar, noting the speciﬁc objective of the

experiment. List and deﬁne all of the independent and dependent vari-

ables, parameters, and measurands. Also provide any equation(s) that

involve the variables and deﬁne each term.

3. Give one historical example of an experiment falling into each of the four

categories of experimental purpose. Describe each experiment brieﬂy.

4. Write a brief description of the very ﬁrst experiment that you ever per-

formed. What was its purpose? What were its variables?

5. What do you consider to be the greatest experiment ever performed?

Explain your choice. You may want to read about the 10 ‘most beauti-

ful experiments of all time’ voted by physicists as reported by George

Johnson in the New York Times on September 24, 2002, in an article

titled “Here They Are, Science’s 10 Most Beautiful Experiments.” Also

see R.P. Crease, 2003. The Prism and the Pendulum: The Ten Most

Beautiful Experiments in Science. New York: Random House.

6. Select one of the 10 most beautiful physics experiments. (See

http://physics-animations.com/Physics/English/top

ref.htm). Brieﬂy ex-

plain the experiment and classify its type. Then list the variables in-

volved in the experiment. Finally, classify each of these variables.

7. Measure the volume of your room and ﬁnd the number of molecules in

it. Is this an experiment? If so, classify it.

8. Classify these types of experiments: (a) measuring the eﬀect of humidity

on the Young’s modulus of a new ‘green’ building material, (b) demon-

strating the eﬀect of the acidity of carbonated soda by dropping a dirty

12 Measurement and Data Analysis for Engineering and Science

penny into it, (c) determining whether a carbon nanotube is stronger

than a spider web thread.

9. Consider an experiment where a researcher is attempting to measure

the thermal conductivity of a copper bar. The researcher applies a heat

input q

, which passes through the copper bar, and four thermocouples

to measure the local bar temperature T (x). The thermal conductivity,

k, can be calculated from the equation

= −k

Variables associated with the experiment are the (a) thermal conductiv-

ity of the bar, (b) heater input, (c) temperature of points 1, 2, 3, and 4

from the thermocouples, (d) pressure and temperature of the surround-

ing air, (e) smoothness of copper bar at the interfaces with the heaters,

and (f) position of the thermocouples. Determine whether each variable

is dependent, independent, or extraneous. Then determine whether each

variable is a parameter or a measurand.

Bibliography

[1] A. Gregory. 2001. Eureka! The Birth of Science. Duxford, Cambridge,

UK: Icon Books.

[2] R. Harr´e. 1984. Great Scientiﬁc Experiments. New York: Oxford Univer-

sity Press.

[3] D.J. Boorstin. 1985. The Discoverers. New York: Vintage Books.

[4] G. Gale. 1979. The Theory of Science. New York: McGraw-Hill.

[5] Coleman, H.W. and W.G. Steele. 1999. Experimentation and Uncertainty

Analysis for Engineers, 2nd ed. New York: Wiley Interscience.

[6] P. Medawar. 1979. Advice to a Young Scientist. New York: Harper and

Row.

[7] R.L. Park. 2000. Voodoo Science: The Road from Foolishness to Fraud.

New York: Oxford University Press.

[8] R. Feynman. 1994. The Character of Physical Law. Modern Library Edi-

tion. New York: Random House.

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Electronics

CONTENTS

2.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Concepts and Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.5 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.6 Resistance and Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.7 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.8 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.9 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.3 Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4 Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.5 Voltage Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.6 Current Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 RLC Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Elementary DC Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Elementary AC Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 *Equivalent Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8 *Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.9 *Impedance Matching and Loading Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.10 *Electrical Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.11 Problem Topic Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.12 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.13 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Nothing is too wonderful to be true, if it be consistent with the laws of nature

and in such things as these, experiment is the best test of such consistency.

Michael Faraday (1791-1867) on March 19, 1849. From a display at the Royal

Institution, London.

... the language of experiment is more authoritative than any reasoning: facts

can destroy our ratiocination − not vice versa.

Alessandro Volta (1745-1827), quoted in The Ambiguous Frog: The Galvani-Volta

Controversy on Animal Electricity, M. Pera, 1992.