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Measurement and
Control Basics, 3rd Edition
Thomas A. Hughes
• Process Control and Process Control Loops
• Fundamentals
• Temperature and Pressure Measurement
• Level Measurement and Control
• Analytical and Flow Measurement
Taken from Measurement and Control Basics, Third Edition

Notice
The information presented in this publication is for the general education of the reader. Because neither the
authors nor the publisher have any control over the use of the information by the reader, both the authors and
the publisher disclaim any and all liability of any kind arising out of such use. The reader is expected to
exercise sound professional judgment in using any of the information presented in a particular application.
Additionally, neither the authors nor the publisher have investigated or considered the effect of any patents on
the ability of the reader to use any of the information in a particular application. The reader is responsible for
reviewing any possible patents that may affect any particular use of the information presented.
Any references to commercial products in the work are cited as examples only. Neither the authors nor the
publisher endorse any referenced commercial product. Any trademarks or tradenames referenced belong to the
respective owner of the mark or name. Neither the authors nor the publisher make any representation regarding
the availability of any referenced commercial product at any time. The manufacturer’s instructions on use of
any commercial product must be followed at all times, even if in conflict with the information in this
publication.
Copyright © 2002 ISA – The Instrumentation, Systems, and Automation Society
All rights reserved.
Printed in the United States of America.
10 9 8 7 6 5 4 3 2
ISBN 1-55617-764-X
No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any
means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of
the publisher.
ISA
67 Alexander Drive
P.O. Box 12277
Research Triangle Park, NC 27709
Library of Congress Cataloging-in-Publication Data
Hughes, Thomas A.
Measurement and control basics / Thomas A. Hughes.-- 3rd ed.
p. cm. -- (Resources for measurement and control series)
Includes bibliographical references and index.
ISBN 1-55617-764-X
1. Process control--Instruments. 2. Measuring instruments. I. Title.
II. Series.
TS156.8 .H78 2001
670.42'7--dc21
2001006083

Editor’s Introduction
This “mini-book” is available both in downloadable form, as part of the ISA Press Digital Book Library,
and bound in a print format.
“Mini-books” are small, unified volumes, from 25 to 100 pages long, drawn from the ISA catalog of
reference and technical books. ISA makes mini-books available to readers who need narrowly focused
information on particular subjects rather than a broad-ranging text that provides an overview of the entire
subject. Each provides the most recent version of the material—in some cases including revisions that have
not yet been incorporated in the larger parent volume. Each has been re-indexed and renumbered so it can
be used independently of the parent volume. Other mini-books on related subjects are available.
The material in this mini-book was drawn from the following ISA titles:
• Measurement and Control Basics, 3rd Edition, by Thomas A. Hughes.
Order Number: 1-55617-764-X
To order: Internet: www.isa.org
Phone: 919/549-8411
Fax: 919/549-8288
Email: info@isa.org

ISA Resources for Measurement and Control Series (RMC)
• Measurement and Control Basics, 3rd Edition (2002)
• Industrial Level, Pressure, and Density Measurement (1995)
• Industrial Flow Measurement (1990)
• Programmable Controllers, 3rd Edition (2001)
• Control Systems Documentation: Applying Symbols and Identification (1993)
• Industrial Data Communications: Fundamentals and Applications,
3rd Edition (2002)
• Real-Time Control Networks (1993)
• Automation Systems for Control and Data Acquisition (1992)
• Control Systems Safety Evaluation and Reliability, 2nd Edition(1998)

THIS BOOK IS DEDICATED TO
my wife Ellen, my daughter Audrey, and my mother Helene
for their love

ix
CONTENTS
ABOUT THE AUTHOR xiii
PREFACE xv
Chapter 1 INTRODUCTION TO PROCESS CONTROL 1
Introduction, 1
Definition of Process Control, 1
Elements of a Process Control System, 3
General Requirements of a Control System, 7
Intuitive Approach to Process Control Concepts, 9
Chapter 2 PROCESS CONTROL LOOPS 27
Introduction, 27
Single-loop Feedback Control, 27
Time Elements of a Feedback Loop, 30
Comparison of Basic Physical Systems, 35
Dead Time Lag, 47
Advanced Control Loops, 49
Tuning Control Loops, 53
Chapter 3 ELECTRICAL AND ELECTRONIC FUNDAMENTALS 67
Introduction, 67
Fundamentals of Electricity, 67
Selecting Wire Size, 81
Electrical Control Devices, 87
Chapter 4 DIGITAL SYSTEM FUNDAMENTALS 93
Introduction, 93
Binary Signals and Codes, 93

x Table of Contents
Numbering Systems, 94
Data Codes, 101
Binary Logic Functions, 106
Logic Function Symbols, 111
Ladder Logic Diagrams, 111
Chapter 5 PRESSURE MEASUREMENT 117
Introduction, 117
Definition of Pressure, 117
Manometers, 126
Pressure Gauges, 128
Chapter 6 LEVEL MEASUREMENT AND CONTROL 147
Introduction, 147
Sight-type Instruments, 147
Pressure-type Instruments, 151
Electrical-type Instruments, 155
Sonic-type Instruments, 160
Radiation-type Instruments, 161
Level Switches, 165
Chapter 7 TEMPERATURE MEASUREMENT 171
Introduction, 171
A Brief History of Temperature Measurement, 171
Temperature Scales, 172
Reference Temperatures, 173
Filled-System Thermometers, 175
Bimetallic Thermometers, 176
Thermocouples, 179
Resistance Temperature Detectors, 188
Thermistors, 193
Integrated-Circuit Temperature Sensors, 195
Radiation Pyrometers, 197
Chapter 8 ANALYTICAL MEASUREMENT AND CONTROL 201
Introduction, 201
Conductivity Measurement, 201
Hydrogen-Ion Concentration (pH) Measurement, 204
Density and Specific Gravity Measurement, 208
Humidity Measurement, 216
Principles of Electromagnetic Radiation, 221
Electromagnetic Spectrum, 221
Photodetectors, 224
Turbidity Analyzer, 231
Gas Analysis, 232
Analyzer Measurement Applications, 236

Table of Contents xi
Chapter 9 FLOW MEASUREMENT 241
Introduction, 241
Flow Principles, 241
Flow-Measuring Techniques, 252
Chapter 10 FINAL CONTROL ELEMENTS 275
Introduction, 275
Control Valve Basics, 275
AC and DC Motors, 292
Pumps, 302
Chapter 11 PROCESS CONTROL COMPUTERS 309
Introduction, 309
History of Process Control Computers, 309
Distributed Control Systems, 315
Programmable Controllers, 318
Basic Components of PLC Systems, 320
Plantwide Computer-based System, 336
Appendix A STANDARD GRAPHICS SYMBOLS FOR PROCESS CONTROL
AND INSTRUMENTATION 341
Appendix B THERMOCOUPLE TABLES 353
Appendix C ANSWERS TO EXERCISES 357
INDEX 367

xiii
ABOUT THE AUTHOR
Thomas A. Hughes, a Senior Member of ISA—The Instrumentation, Sys-
tems, and Automation Society, has 30 years of experience in the design
and installation of instrumentation and control systems, including 20
years in the management of instrumentation and control projects for the
process and nuclear industries. He is the author of two books: Measure-
ment and Control Basics, 3rd
Edition, (2002) and Programmable Controllers, 3rd
Edition, (2001), both published by ISA.
Mr. Hughes received a B. S. in engineering physics from the University of
Colorado, and a M.S. in control systems engineering from Colorado State
University. He holds professional engineering licenses in the states of Col-
orado and Alaska, and has held engineering and management positions
with Dow Chemical, Rockwell International, EG&G Rocky Flats, Topro
Systems Integration, and the International Atomic Energy Agency. Mr.
Hughes has taught numerous courses in electronics, mathematics, and
instrumentation systems at the college level and in industry. He is cur-
rently the Principal Consultant with Nova Systems Engineering Services
in Arvada, Colorado.

xv
Preface
The third edition of Measurement and Control Basics is a thorough and com-
prehensive treatment of the basic principles of process control and mea-
surement. It is designed for engineers, technicians, management, and sales
personnel who are new to process control and measurement. It is also
valuable as a concise and easy-to-read reference source on the subject.
This new edition provides expanded coverage of pressure, level, flow,
temperature, analytical measurement, and process control computers.
Material on the proper tuning of control loops was added to Chapter 1,
and expanded coverage of control loops was added to Chapter 2. Chapter
3 includes a more complete discussion of electrical and electronic funda-
mentals needed in process control and instrumentation.
The discussion of the basic principles underlying pressure measurement
has been expanded to include a discussion of sensor characteristics and
potentiometric-type pressure sensors. Extensive coverage was added on
typical pressure transmitter applications. The discussion on level mea-
surement has been increased with the addition of several common level
instruments and switches such as displacers, tape floats, microwave, and
radar. The chapter on temperature measurement has been improved by
adding new illustrations and a section on radiation pyrometers. Coverage
of analytical measurement and control in Chapter 8 was increased by the
addition of a section on the principles of electromagnetic radiation and its
application to analytical measurement. Three sections were also added to
Chapter 8 on photoconductive sensors, photomultiplier tubes, and turbid-
ity analyzers.
Chapter 9 on flow measurement contains new coverage on Reynolds
Number and fluid flow profiles. The discussion of the basic principles of

xvi Measurement and Control Basics
fluid flow has been expanded and improved in Chapter 9. A discussion on
types of control valves and control valve actuators was added to Chapter
10 and the section on control valve sizing was expanded and improved.
All of the chapters have been supplemented with new or improved exam-
ple problems and exercises. Most of the illustrations in the book have been
revised and improved.

1
1
Introduction to
Process Control
Introduction
To study the subject of industrial process control effectively you must first
gain a general understanding of its basic principles. To present these con-
trol principles clearly and concisely, an intuitive approach to process con-
trol is used. First, however, some basic definitions and concepts of process
control are presented.
Definition of Process Control
The operations that are associated with process control have always
existed in nature. Such “natural” process control can be defined as any
operation that regulates some internal physical characteristic that is
important to a living organism. Examples of natural regulation in humans
include body temperature, blood pressure, and heart rate.
Early humans found it necessary to regulate some of their external envi-
ronmental parameters to maintain life. This regulation could be defined as
“artificial process control” or more simply as “process control,” as we will
refer to it in this book. This type of process control is accomplished by
observing a parameter, comparing it to some desired value, and initiating
a control action to bring the parameter as close as possible to the desired
value. One of the first examples of such control was early man’s use of fire
to maintain the temperature of their environment.
The term automatic process control came into wide use when people learned
to adapt automatic regulatory procedures to manufacture products or pro-

2 Measurement and Control Basics
cess material more efficiently. Such procedures are called automatic
because no human (manual) intervention is required to regulate them.
All process systems consist of three main factors or terms: the manipu-
lated variables, disturbances, and the controlled variables (Figure 1-1).
Typical manipulated variables are valve position, motor speed, damper
position, or blade pitch. The controlled variables are those conditions,
such as temperature, level, position, pressure, pH, density, moisture con-
tent, weight, and speed, that must be maintained at some desired value.
For each controlled variable there is an associated manipulated variable.
The control system must adjust the manipulated variables so the desired
value or “set point” of the controlled variable is maintained despite any
disturbances.
Disturbances enter or affect the process and tend to drive the controlled
variables away from their desired value or set point condition. Typical dis-
turbances include changes in ambient temperature, in demand for prod-
uct, or in the supply of feed material. The control system must adjust the
manipulated variable so the set point value of the controlled variable is
maintained despite the disturbances. If the set point is changed, the
manipulated quantity must be changed to adjust the controlled variable to
its new desired value.
For each controlled variable the control system operators select a manipu-
lated variable that can be paired with the controlled variable. Often the
choice is obvious, such as manipulating the flow of fuel to a home furnace
to control the temperature of the house. Sometimes the choice is not so
obvious and can only be determined by someone who understands the
process under control. The pairing of manipulated and controlled vari-
ables is performed as part of the process design.
Figure 1-1. Process control variables
Process
Disturbances
Controlled
Manipulated
Variables
Variables

Chapter 1 – Introduction to Process Control 3
Elements of a Process Control System
Figure 1-2 illustrates the essential elements of a process control system. In
the system shown, a level transmitter (LT), a level controller (LC), and a
control valve (LV) are used to control the liquid level in a process tank. The
purpose of this control system is to maintain the liquid level at some pre-
scribed height (H) above the bottom of the tank. It is assumed that the rate
of flow into the tank is random. The level transmitter is a device that mea-
sures the fluid level in the tank and converts it into a useful measurement
signal, which is sent to a level controller. The level controller evaluates the
measurement, compares it with a desired set point (SP), and produces a
series of corrective actions that are sent to the control valve. The valve con-
trols the flow of fluid in the outlet pipe to maintain a level in the tank.
Thus, a process control system consists of four essential elements: process,
measurement, evaluation, and control. A block diagram of these elements is
shown in Figure 1-3. The diagram also shows the disturbances that enter
or affect the process. If there were no upsets to a process, there would be
no need for the control system. Figure 1-3 also shows the input and output
of the process and the set point used for control.
Figure 1-2. Process level control: Example
Liquid
H
LT
100
Control
Valve
Liquid
LC
100
LV
100
Level
Transmitter
Level
Controller

Process
In general, a process consists of an assembly of equipment and material
that is related to some manufacturing operation or sequence. In the exam-
ple presented in Figure 1-2, the process whose liquid level is placed under
control includes such components as a tank, the liquid in the tank, the flow
of liquid into and out of the tank, and the inlet and outlet piping. Any
given process can involve many dynamic variables, and it may be desir-
able to control all of them. In most cases, however, controlling only one
variable will be sufficient to control the process to within acceptable limits.
One occasionally encounters a multivariable process in which many vari-
ables, some interrelated, require regulation.
Measurement
To control a dynamic variable in a process, you must have information
about the entity or variable itself. This information is obtained by measur-
ing the variable.
Measurement refers to the conversion of the process variable into an ana-
log or digital signal that can be used by the control system. The device
that performs the initial measurement is called a sensor or instrument. Typ-
ical measurements are pressure, level, temperature, flow, position, and
speed. The result of any measurement is the conversion of a dynamic vari-
able into some proportional information that is required by the other ele-
ments in the process control loop or sequence.
Evaluation
In the evaluation step of the process control sequence, the measurement
value is examined, compared with the desired value or set point, and the
amount of corrective action needed to maintain proper control is deter-
Figure 1-3. Four elements of a control system
Measurement
Control
Evaluation
Input
Set Point
Process
Disturbances
Output

mined. A device called a controller performs this evaluation. The controller
can be a pneumatic, electronic, or mechanical device mounted in a control
panel or on the process equipment. It can also be part of a computer con-
trol system, in which case the control function is performed by software.
Control
The control element in a control loop is the device that exerts a direct influ-
ence on the process or manufacturing sequence. This final control element
accepts an input from the controller and transforms it into some propor-
tional operation that is performed on the process. In most cases, this final
control element will be a control valve that adjusts the flow of fluid in a
process. Devices such as electrical motors, pumps, and dampers are also
used as control elements.
Process and Instrumentation Drawings
In the measurement and control field, a standard set of symbols is used to
prepare drawings of control systems and processes. The symbols used in
these drawings are based on the standard ANSI/ISA-5.1-1984 (R1992)
Instrumentation Symbols and Identification, which was developed by
ISA—The Instrumentation, Systems, and Automation Society (ISA) and
the American National Standards Institute (ANSI). A typical application
for this standard are process and instrumentation diagrams (P&IDs),
which show the interconnection of the process equipment and the instru-
mentation used to control the process. A portion of a typical P&ID is
shown in Figure 1-4.
In standard P&IDs, the process flow lines, such as process fluid and steam,
are indicated with heavier solid lines than the lines that are used to repre-
sent the instrument. The instrument signal lines use special markings to
indicate whether the signal is pneumatic, electric, hydraulic, and so on.
Table A-1 in appendix A lists the instrument line symbols that are used on
P&IDs and other instrumentation and control drawings. In Figure 1-4, two
types of instrument signals are used: double cross-hatched lines denote
the pneumatic signals to the steam control valve and the process outlet
flow control valve, and a dashed line is used for the electrical control lines
between various instruments. In process control applications, pneumatic
signals are almost always 3 to 15 psig (i.e., pounds per square inch, gauge
pressure), and the electric signals are normally 4 to 20 mA (milliamperes)
DC (direct current).
A balloon symbol with an enclosed letter and number code is used to rep-
resent the instrumentation associated with the process control loop. This

letter and number combination is called an instrument identification or
instrument tag number.
The first letter of the tag number is normally chosen so that it indicates the
measured variable of the control loop. In the sample P&ID shown in Fig-
ure 1-4, T is the first letter in the tag number that is used for the instru-
ments in the temperature control loop. The succeeding letters are used to
represent a readout or passive function or an output function, or the letter
can be used as a modifier. For example, the balloon in Figure 1-4 marked
TE represents a temperature element and that marked TIC is a tempera-
ture-indicating controller. The line across the center of the TIC balloon
symbol indicates that the controller is mounted on the front of a main con-
trol panel. No line indicates a field-mounted instrument, and two lines
means that the instrument is mounted in a local or field-mounted panel.
Dashed lines indicate that the instrument is mounted inside the panel.
Normally, sequences of three- or four-digit numbers are used to identify
each loop. In our process example (Figure 1-4), we used loop numbers 100
and 101. Smaller processes use three-digit loop numbers; larger processes
or complex manufacturing plants may require four or more digits to iden-
tify all the control loops.
Special marks or graphics are used to represent process equipment and
instruments. For example, in our P&ID example in Figure 1-4 two parallel
Figure 1-4. P&ID: Example
Orifice
FE
101
FT FIC
101
101
Control
Heated
FV
101
TV
100
TT
100
TE
100
TIC
100
Steam
Steam
Manual
Fluid
Value
Fluid
Plate
Condensate
Valve

lines represent the orifice plate that is used to detect the discharge flow
from the process heater. The two control valves in the figure also use a spe-
cial symbol. See appendix A for a more detailed discussion of the instru-
mentation and process symbols that are used on P&IDs.
General Requirements of a Control System
The primary requirement of a control system is that it be reasonably stable.
In other words, its speed of response must be fairly fast, and this response
must show reasonable damping. A control system must also be able to
reduce the system error to zero or to a value near zero.
System Error
The system error is the difference between the value of the controlled vari-
able set point and the value of the process variable maintained by the sys-
tem. The system error is expressed in equation form by the following:
e(t) = PV(t) – SP(t) (1-1)
where
e(t) = system error as a function of time (t)
PV(t) = the process variable as a function of time
SP(t) = is the set point as a function of time
System Response
The main purpose of a control loop is to maintain some dynamic process
variable (pressure, flow, temperature, level, etc.) at a prescribed operating
point or set point. System response is the ability of a control loop to
recover from a disturbance that causes a change in the controlled process
variable.
There are two general types of good response: underdamped (cyclic
response) and damped. Figure 1-5 shows an underdamped or cyclic
response of a system in which the process variable oscillates around the
set point after a process disturbance. The wavy response line shown in the
figure represents an acceptable response if the process disturbance or
change in set point was large, but it would not be an acceptable response if
the change from the set point was small.
Figure 1-6 shows a damped response where the control system is able to
bring the process variable back to the operating point with no oscillations.

Control Loop Design Criteria
Many criteria are employed to evaluate the process control’s loop response
to an input change. The most common of these include settling time, max-
imum error, offset error, and error area (Figure 1-7).
Figure 1-5. Cyclic response to process disturbance
Figure 1-6. Damped response to process disturbance
Figure 1-7. Evaluation of control loop response
PV (Process Variable)
Disturbance
Time
Set Point
Disturbance
Time
Set Point
Time
Set Point
Max. Error
Offset
Error
Error Areas
Settling Time

When there is a process disturbance or a change in set point, the settling
time is defined as the time the process control loop needs to bring the pro-
cess variable back to within an allowable error. The maximum error is sim-
ply the maximum allowable deviation of the dynamic variable. Most
control loops have certain inherent linear and nonlinear qualities that pre-
vent the system from returning the process variable to the set point after a
system change. This condition is generally called “offset error” and will be
discussed later in this chapter. The error area is defined as the area between
the response curve and the set point line as shown by the shaded area in
Figure 1-7.
These four evaluation criteria are general measures of control loop behav-
ior that are used to determine the adequacy of the loop’s ability to perform
some desired function. However, perhaps the best way to gain a clear
understanding of process control is to take an intuitive approach.
Intuitive Approach to Process Control Concepts
The practice of process control arose long before the theory or analytical
methods underlying it were developed. Processes and controllers were
designed using empirical methods that were based on intuition (“feel”)
and extensive process experience. Most of the reasoning involved was
nonmathematical. This approach was unscientific trial and error, but it
was a successful control method.
Consider, for example, an operator looking into an early metal processing
furnace to determine whether the product was finished. He or she used
flame color, amount of smoke, and process time to make this judgment.
From equally direct early methods evolved most of the control concepts
and hardware used today. Only later did theories and mathematical tech-
niques emerge to explain how and why the systems responded as they
did.
In this section, we will approach the study of control fundamentals in
much the same way that control knowledge developed—that is, through a
step-by-step procedure starting from manual control and moving to ever-
increasing automatic control.
Suppose we have a process like that shown in Figure 1-8. A source of feed
liquid flows into a tank at a varying rate from somewhere else in a process
plant. This liquid must be heated so that it emerges at a desired tempera-
ture, Td, as a hot liquid. To accomplish this, hot water, which is available
from another part of the plant, flows through heat exchanger coils in the
tank. By controlling the flow of hot water, we can obtain the desired tem-

perature, Td. A further process requirement is that the level of the tank
must neither overflow nor fall so low that it exposes the heater coils.
The temperature is measured in the tank, and a temperature transmitter
(TT-1) converts the signal into a 4-20 mA direct current (DC) signal to
drive a temperature indicator (TI-1) mounted near the hot water inlet
valve. Similarly, a level indicator (LI-2) is mounted within the operator's
view of the hot feed outlet valve (HV-2).
Suppose a process operator has the task of holding the temperature, T,
near the desired temperature, Td, while making sure the tank doesn't over-
flow or the level get too low. The question is how the operator would cope
with this task over a period of time. He or she would manually adjust the
hot water inlet valve (HV-1) to maintain the temperature and occasionally
adjust the outlet valve (HV-2) to maintain the correct level in the tank.
The operator would face several problems, however. Both indicators
would have to be within the operator's view, and the manual valves
would have to be close to the operator and easy to adjust.
On/Off Control
To make the operator's work easier, suppose we installed electrically oper-
ated solenoid valves in place of the manual valves, as shown in Figure 1-9.
We can also install two hand switches (HS-1 and HS-2) so the solenoid
Figure 1-8. Example process – using manual valves
Hot
Water
Manual
Valve Process
Tank
Feed
Manual
Valve
TI
1 1
TT
LT
2 2
LI
Water
Hot
Feed

valves can be operated from a common location. The valves can assume
two states, either fully open (on) or fully closed (off). This type of control is
called two-position or on/off control.
Assume for the moment that the level is holding steady and that the main
concern is controlling temperature. The operator has been told to keep the
temperature of the fluid in the tank at 100°F. He or she compares the read-
ing of the temperature indicator with the selected set point of 100°F. The
operator closes the hot water valve when the temperature of the fluid in
the tank rises above the set point (Figure 1-10). Because of process dead
time and lags the temperature will continue to rise before reversing and
moving toward the set point. When the temperature falls below 100°F, the
operator opens the hot water valve. Again, dead time and lags in the pro-
cess create a delay before the temperature begins to rise. As it crosses the
set point, the operator again shuts off the hot water, and the cycle repeats.
This cycling is normal for a control system that uses on/off control. This
limitation exists because its impossible for the operator to control the pro-
cess exactly with only two options.
This on/off type of control can be expressed mathematically as follows:
e = PV – SP (1-2)
Figure 1-9. Sample process: Solenoid valves
Hot
Valve
FV
1
Process
Tank
Valve
2
TI
1 1
TT
LT
2 2
LI
Hot
S
S
Valve
1
Process
Tank
Feed
Valve
2
TI
1 1
TT
LT
2 2
LI
S
S
HS
2
HS
1
FV
Solenoid
Solenoid
Feed
Water
Water

where
e = the error
SP = the set point
PV = the process variable
In the on/off control mode, the valve is open valve when the error (e) is
positive (+), and the valve is closed when e is negative (–).
Proportional Control
When we view the process as a balance between energy in and energy out,
it is clear that smoother control would result if a steady flow of hot water
were maintained rather than the sudden changes between ON and OFF.
The problem is finding the correct value for the steady flow required for
proper control. Obviously, for each rate of feed flow in and out of the tank,
some ideal amount of inlet water flow exists that will hold the outlet tem-
perature, T, at 100°F.
This suggests that we should make two modifications to our control mode
or strategy. The first is to establish some steady-flow value for the hot
water that, at average operating conditions, tends to hold the process vari-
able (temperature) at the desired value or set point (100°F). Once that aver-
age flow value has been established for the hot water, increases or
decreases of error (e = SP – PV) must be allowed to cause corresponding
increases and decreases in water flow from this normal value. This illus-
trates the concept of proportional control (i.e., initiating a corrective action to
a value that is in some proportion to the change in error or deviation of the
process variable from set point).
Figure 1-10. “On/off” temperature control
100oF
Temperature Changes
Set Point (SP)
Open
Valve Position (V)
Closed
Process Variable (PV)

Before proportional control can be implemented on our sample process,
we must change the solenoid valves to adjustable control valves. Such
valves can be positioned to any degree of opening—from fully closed to
fully opened—depending on the type of valve actuator mechanism you
choose (generally either an electrically or pneumatically operated dia-
phragm actuator). Our sample process now looks like Figure 1-11, which
now shows the use of pneumatically operated control valves (TV-1 and
LV-2) and process controllers (TIC-1 and LIC-2). Control valves and
controllers in the system make it possible to achieve better control of the
process.
Proportional control can be described mathematically as follows:
V = Kce + m (1-3)
where
V = is the control valve position
Kc = is the adjustable proportional gain of a typical process
controller
m = is a constant, which is the position of the control valve
when the system error (e) is zero.
Proportional control can be illustrated by using the three graphs in
Figure 1-12 and setting the proportional constant to three different values
(i.e., Kc = 1, Kc <1, and Kc >1).
Figure 1-11. Sample process: Proportional control
Hot
Water
Process
Tank
Hot Feed
LT
2 2
LIC
Water
Water
TV
1
1
Process
Tank
1
TT
2 2
Raw Feed
LY
2
TY
1
LV
2
TY
1
IA
IA
I/P

As these graphs show, the amount of valve change (∆V) for a given error
can vary substantially. A one-to-one relationship is shown in Figure
1-12(a). In that example, the control valve would move 1 percent of its full
travel for a corresponding 1 percent change in error or in a one-to-one
Figure 1-12. Proportional control
Valve Position (V)
Open
100%
Close
0% SP
e
error
m
a) Gain of one, KC = 1
Open
Close
0% SP 100%
m
e
V
b) Low Gain, KC < 1.
Open
Close
c) High Gain, KC > 1.
0% SP 100%
e
V
error
V
PV
PV
PV
Valve Position (V)
Valve Position (V)

ratio. In Figure 1-12(b), where a low gain (Kc <1) is selected, a large change
in error is required before the control valve would be fully opened or
closed. Finally, Figure 1-12(c) shows the case of high gain (Kc >1), where a
very small error would cause a large change in the control valve position.
The term proportional gain, or simply gain, arose as a result of the use of
analytical methods in process control. Historically, the proportionality
between error and valve action was called proportional band (PB). Propor-
tional band is the expression that states the percentage of change in error
that is required to move the valve full scale. Again, this had intuitive plau-
sibility because it gave an operator a feel for how small of an error caused
full corrective action. Thus, a 10 percent proportional band meant that a 10
percent error between SP and PV would cause the output to go full scale.
This definition can be related to proportional gain Kc by noting the follow-
ing equation:
(1-4)
An example will help you understand the relationship between propor-
tional band and gain.
The modern way of considering proportional control is to think in terms of
gain (Kc). The m term, as Equation 1-3 shows, has to be that control valve
position that supplies just the right amount of hot water to make the tem-
perature 100°F, that is, PV = SP. The position, m, indicated in Figures
1-12(a), (b), and (c), is often called the manual (m) reset because it is a man-
ual controller adjustment.
EXAMPLE 1-1
Problem: For a proportional process controller:
What proportional band corresponds to a gain of 0.4?
What gain corresponds to a PB of 400?
Solution: a)
b)
1
x 100
C
PB
K
=
1 100
x 100 = = 250%
0.4
C
PB
K
=
1 100
x 100 = 0.25
400
C
K
PB
= =

When a controller is designed to provide this mode of control, it must con-
tain at least two adjustments: one for the Kc term and one for the m term.
Control has become more complicated because it is now necessary to
know where to set Kc and m for best control.
It would not take too long for the operator of our sample process to dis-
cover a serious problem with proportional control, namely, proportional
control rarely ever keeps the process variable at the set point if there are
frequent disturbances to the process. For example, suppose the flow to the
tank suddenly increases. If the temperature of the tank is to be maintained
at 100°F at this new rate of feed flow, more hot water must be supplied.
This calls for a change in valve position. According to Equation 1-3, the
only way that the valve position (V) can be changed is for the error (e) to
change. Remember that m is a constant. Thus, an error will occur, and the
temperature will drop below 100°F until an equilibrium is reached
between the hot water flow and new feed flow. How much this drop will
be depends on the value of Kc that was set in the controller as well as on
the characteristics of the process. The larger Kc is, the smaller this offset
will be in a given system.
However, it can be shown that Kc cannot be increased indefinitely because
the control loop will become unstable. So, some error is inevitable if the
feed rate changes. These points are illustrated in Figure 1-13, which shows
a plot of hot feed temperature versus hot water flow rate (valve position)
for both low raw feed flow and high raw feed flow.
Figure 1-13. Sample process: Temperature vs. valve position
Hot Feed
Temperature
T1
T2
Hot Water Valve Position
Position 1 Position 2
Low Inlet
Feed Flow
High Inlet
Feed Flow

For the hot water valve in position 1 and the raw feed coming into the pro-
cess tank at the low flow rate, the process would heat the fluid and pro-
duce hot feed fluid at temperature T2. If suddenly the feed went to the
high flow rate and the valve position was not changed, the temperature
would drop to T1. At this new high flow rate, the hot water valve must be
moved to position 2 if the original temperature T2 is to be restored. Figure
1-14 shows the extent to which proportional control of the temperature
valve can achieve this restoration.
One way to cope with the offset problem is by manually adjusting the m
term. When we adjust the m term (usually through a knob on a process
controller), we are moving the valve to a new position that allows PV to
equal SP under the new conditions of load. In this case, with an increase in
feed flow, Equation 1-3 clearly shows that the only way to obtain a new
value for V, if e is to be zero, is by changing the m term. If process changes
are frequent or large it may become necessary to adjust m frequently. It is
apparent that some different type of control mode is needed.
Proportional-Plus-Integral Control
Suppose that the controller rather than the operator manually adjusts the
proportional controller described in the previous section. This would elim-
inate the offset error caused by process changes. The question then is, on
what basis should the manual reset be automated? One innovative con-
cept would be to move the valve at some rate, as long as the error is not
zero. Though eventually the correct control valve position would be
found, there are many rates at which to move the valve. The most com-
mon practice in the instrumentation field is to design controllers that
move the control valve at a speed or rate proportional to the error. This has
Figure 1-14. Process response with proportional control
Process Variable
Inlet Feed Flow
Time
Time
Step Change
Offset
Set Point
No Control Action
With proportional control

some logic to it, in that it would seem plausible to move the valve faster as
the error got larger. This added control mode is called reset or integral
action. It is usually used in conjunction with proportional control because
it eliminates the offset.
This proportional-plus-integral (PI) control is shown in Figure 1-15. Assume a
step change in set point at some point in time, as shown in the figure. First,
there is a sudden change in valve position equal to Kce due to the propor-
tional control action. At the same time, the reset portion of the controller,
sensing an error, begins to move the valve at a rate proportional to the
error over time. Since the example in Figure 1-15 had a constant error, the
correction rate was constant.
When time is used to express integral or reset action, it is called the reset
time. Quite commonly, its reciprocal is used, in which case it is called reset
rate in “repeats per minute.” This term refers to the number of times per
minute that the reset action is repeating the valve change produced by
proportional control alone. Process control systems personnel refer to reset
time as the integral time and denote it as ti.
The improvement in control that is caused by adding the integral or reset
function is illustrated in Figure 1-16. The same process change is used that
was previously assumed under proportional-only control. Now, however,
Figure 1-15. Proportional-plus-integral control
valve
position
Error (e)
Time (t)
No error (SP = PV)
0
Time (t)
Proportional Contribution
+ e
- e
Reset Contribution
KCe
KCe
ti
Integral
or Reset
time
Open
Closed

Figure 1-16. Process response with PI control
Valve Position
V
Open
100%
Closed
0% SP
e
error
m
a) Gain of one, KC = 1
V
Open
Closed
0% SP 100%
m
e
V
b) Low Gain, KC < 1.
V
Open
Closed
c) High Gain, KC > 1.
0% SP 100%
e
Valve Position
Valve Position
V
Error
V
Process Variable
Range (0 to 100%)
Process Variable
Range (0 to 100%)
Process Variable
Range (0 to 100%)

after the initial upset the reset action returns the error to zero and there is
no offset.
Recognizing that the reset action moves the control valve at a rate propor-
tional to error, this control mode is described mathematically as follows:
(1-5)
where
dV/dt = the derivative of the valve position with respect to time (t)
Ki = an adjustable constant
We can find the position of the valve at any time by integrating this differ-
ential equation (Equation 1-5). If we integrate from time 0 to time, t, we
obtain:
(1-6)
This equation shows that the control valve position is proportional to the
integral of the error. This fact leads to the “integral control” label. Finally,
combining proportional and integral control gives the total expression of a
two-mode proportional-plus-integral (PI) controller:
(1-7)
If we let Ki = Kc/ti, we obtain an alternate form of the PI control equation
in terms of the proportional constant, Kc, and the integral time, ti, as fol-
lows:
(1-8)
One problem with PI control bears mentioning. If a control loop is using PI
control, the possibility exists with the integral (reset) mode that the con-
troller will continue to integrate and change the output even outside the
operating range of the controller. This condition is called “reset windup.”
For example, the heat exchanger shown in Figure 1-17 can be designed
and built to heat 50 gal/min of process fluid from 70°F to 140°F. If the pro-
cess flow should suddenly increase to 100 gal/min, it may be impossible
to supply sufficient steam to maintain the process fluid temperature at
140°F even when the control valve is wide open (100%), as shown in Fig-
i
dV
K e
dt
=
0
t
i
V K edt
= ∫
0
t
c i
V K e K edt
= + ∫
0
t
c
c
i
K
V K e edt
t
= + ∫

ure 1-18. In this case, the reset mode, having opened the valve all the way
(the controller output is perhaps 15 psig), would continue to integrate the
error signal and increase the controller output all the way in order to sup-
ply pressure from the pneumatic system. Once past 15 psig, the valve will
open no further, and the continued integration serves no purpose. The
controller has “wound up” to a maximum output value.
Further, if the process flow should then drop to 50 gal/min (back to the
operable range of the process), there would be a period of time during
which the controlled temperature is above the set point while the valve
remains wide open. It takes some time for the integral mode to integrate
(reset) downward from this wound-up condition to 15 psig before the
valve begins to close and control the process.
It is possible to prevent this problem of controller-reset windup by using a
controller operational feature that limits the integration and the controller
output. This feature is normally called anti-reset windup and is recom-
mended for processes that may periodically operate outside their capacity.
Proportional-Plus-Derivative (PD) Control
We can now add to proportional control another control action called
derivative action. This control function produces a corrective action that is
proportional to the rate of change of error. Note that this additional correc-
tion exists only while the error is changing; it disappears when the error
stops changing, even though there may still be a large error.
Figure 1-17. Heat exchanger temperature control
TIC
200
IA TY
200
Steam
I/P
Process
Fluid
Process
Fluid
Condensate
TT
200
TE
200
TV
200
E-200

Derivative control can be expressed mathematically as follows:
(1-9)
where
Kd = the derivative constant
de/dt = the derivative of the control system error with respect to
time
The derivative constant Kd can be related to the proportional constant Kc
by the following equation:
(1-10)
Figure 1-18. Reset windup control
Temperature
SP (1400F)
Time (t)
Inlet Feed
Flow (GPM)
Step Change
100
50
100
75
Time (t)
Time (t)
Controller
Output (%)
windup
d
de
V K
dt
=
d c d
K K t
=

where td is the derivative control constant. If we add derivative control to
proportional control, we obtain
(1-11)
To illustrate the effects of PD control, let's assume that the error is chang-
ing at a constant rate. This can be obtained by changing the set point at a
constant rate (i.e., SP = ct), as shown in Figure 1-19.
Derivative action contributes an immediate valve change that is propor-
tional to the rate of change of the error. In Figure 1-19, it is equal to the
slope of the set point line. As the error increases, the proportional action
contributes additional control valve movement. Later, the contribution of
the proportional action will have equaled the initial contribution of the
rate action. The time it takes for this to happen is called the derivative time,
td. The ramped error can be expressed mathematically as follows:
(1-12)
Figure 1-19. Proportional-plus-derivative
c c d
de
V K e K t
dt
= +
Time(t)
Valve
Position
Set Point
e = ct
td
Derivative
Contribution
Proportional
Contribution
Time(t)
e Ct
=

where
e = the control loop error
C = a constant (slope of set point change)
t = time
If we substitute this value (e = Ct) for the control loop error into the equa-
tion for a PD controller, we obtain
(1-13)
Since the derivative of Ct with respect to time, t, is simply equal to C, the
control action (V) from the PD controller to the control valve becomes
(1-14)
This indicates that the valve position is ahead in time by the amount td
from the value that straight proportional control would have established
for the same error. The control action leads to improved control in many
applications, particularly in temperature control loops where the rate of
change of the error is very important. In temperature loops, large time
delays generally occur between the application of corrective action and the
process response; therefore, derivative action is required to control steep
temperature changes.
Proportional-Integral-Derivative Control
Finally, the three control functions—proportional, integral, and derivative
—can be combined to obtain full three-mode or PID control:
(1-15)
Deciding which control action (i.e., PD, PID, etc.) should be used in a con-
trol system will depend on the characteristics of the process being con-
trolled. Three-mode control (PID) cannot be used on a noisy measurement
process or on one that experiences stepwise changes because the deriva-
tive contribution is based on the measurement of rate of change. The
derivative of a true step change is infinite, and the derivatives of a noisy
measurement signal will be very large and lead to unstable control.
The PID controller is used on processes that respond slowly and have long
periods. Temperature control is a common example of PID control because
the heat rate may have to change rapidly when the temperature measure-
( )
c c d
d Ct
V K Ct K t
dt
= +
( )
c d
V K C t t
= +
0
t
c
c c d
i
K de
V K e edt K t
t dt
= + +
∫

ment begins to change. The derivative action shortens the response of the
slow process to an upset.
In the next chapter we discuss such important characteristics of processes
as time constants and dead time. By understanding these concepts you
will be better able to select the proper control action type for effective con-
trol.
EXERCISES
1.1 What are the three main factors or terms found in all process
control systems? List examples of each type.
1.2 List the four essential elements of a process control system.
1.3 What function is performed by a process controller in a control
loop?
1.4 What type of instrument is identified by each of the following
instrument tag numbers?: (a) PIC-200, (b) FV-250, (c) LC-500, and
(d) HS-100.
1.5 What is the primary requirement of any process control system?
1.6 Define the term system error with respect to a control system.
1.7 For a proportional controller, (a) what gain corresponds to a
proportional band of 150 percent and (b) what proportional band
corresponds to a gain of 0.2?
1.8 What is the main reason to use integral action with proportional
control?
1.9 Explain the concept of “reset windup” encountered in
proportional-plus-integral controllers.
1.10 What type of controller is used on the heat exchanger shown in
Figure 1-17? Where is the controller located?
1.11 Discuss the type of process that can most benefit from the use of
PID control.
BIBLIOGRAPHY
1. Chemical Engineering magazine staff. Practical Process Instrumentation and
Control, New York: McGraw-Hill, 1980.
2. Considine, D. M. (ed.). Process Industrial Instruments and Controls Handbook, 4th
ed., New York: McGraw-Hill, 1983.
3. Honeywell International, Process Management Systems Division. An
Evolutionary Look at Process Control, Honeywell International, 1981.

4. John, C. D. Process Control Instrumentation Technology, 2d ed., New York: John
Wiley & Sons, 1982.
5. Kirk, F. W., and N. R. Rimboi. Instrumentation, 3d ed., Homewood, IL:
American Technical Publishers, 1975.
6. Liptak, B. G. (ed.). Process Control Instrument Engineers' Handbook, rev. ed.,
Radnor, PA: Chilton Book, 1985.
7. Murrill, P. W. Fundamentals of Process Control Theory, 3d ed., Research Triangle
Park, NC: ISA, 2000.
8. Ogata, K., Modern Control Engineering, Englewood Cliffs, NJ: Prentice-Hall,
1970.
9. Weyrick, R. C., Fundamentals of Automatic Control, New York: McGraw-Hill,
1975.

27
2
Process Control
Loops
Introduction
We discussed the general concepts of process control in Chapter 1. In this
chapter, we will cover the basic principles of process control loops. Single-
loop feedback control is the most common type of control used in indus-
trial processes, so it will be discussed in the greatest detail. We will then
discuss other types of control loops, such as cascade, ratio, and feedforward.
Finally, we will examine several common methods used to tune control
loops.
Single-loop Feedback Control
In a feedback control loop, the variable to be manipulated is measured.
This measured process value (PV) is then compared with a set point (SP)
to generate an error signal (e = PV - SP). If a difference or error exists
between the actual value and the desired value of the process, a process
controller will take the necessary corrective action to return the process to
the desired value. A block diagram of a single-feedback control loop is
shown in Figure 2-1.
The measured process variable is sensed or measured by the appropriate
instrumentation, such as temperature, flow, level, or analytical sensors.
This measured value is then compared with the set point. The controller
uses this comparison to adjust the manipulated variable appropriately by
generating an output signal. The output signal is based in turn on which-
ever control strategy or algorithm has been selected. Because in the pro-
cess industries the manipulated variable is most often a flow, the output of

the controller is usually a signal to a flow control valve, as shown in
Figure 2-1.
During the operation of the process, disturbances can enter the process
and drive the process variable in one direction or another. The single
manipulated variable is used to compensate for all such process changes
produced by the disturbances. Furthermore, if changes occur in the set
point, the manipulated variable is altered to produce the needed change in
the process output.
Process Controllers
The most dynamic device in a feedback control loop is the process control-
ler. There are three types of controllers—mechanical, pneumatic, and elec-
tronic—and they all serve the same function. They compare the process
variable with the set point and generate an output signal that manipulates
the process to make the process variable equal to its set point. Figure 2-2
shows a block diagram of a feedback control loop with an expanded view
of its common functions. In this diagram the measurement transducer has
been expanded into its two components: the sensor and the transmitter.
The sensor measures the process variable, and then the transmitter con-
verts the measurement into a standard signal such as 4 to 20 mA DC or 3
to 15 psig.
The controller consists of a feedback transmission system, a comparator
with a set point input, controller functions, and an output transmission
system. The comparator block measures the difference between the set
point and the process variable. For this comparison to be useful, the set
point and the process variable must have the same units of measure. For
example, if the set point has the units of 0 to 10 mv, then the signal from
Figure 2-1. Feedback control loop
Control
Valve
Manipulated
Variable
Controller Transmitter
Process
Set Point
Sensor
Controlled
Variable

Chapter 2 – Process Control Loops 29
the sensor must be converted into the same units. The purpose of the feed-
back transmission system is to convert the sensor signal into the correct
units. For example, if the input signal is 4 to 20 mA DC the feedback circuit
in the controller will convert the signal to 0 to 10 mv. The function of the
output transmission system is to convert the signal from the feedback cir-
cuit into the form required by the final control device. The four common
controller functions are proportional, proportional plus integral (PI), pro-
portional plus derivative (PD), and proportional plus integral plus deriva-
tive (PID).
A front-panel view of a typical electronic process controller is shown in
Figure 2-3. The controller has two vertical bar displays to give the operator
a pictorial view of the process variable and the set point. It also has two
short horizontal digital displays just above the vertical bars to give the
operator a direct digital readout of the process variable and the set point.
The operator uses dual push buttons with indicating arrows to adjust the
set point and the manual output functions. The operator must depress the
manual (“M”) push button to activate the manual output function.
During normal operation, the operator will select automatic (“A”) mode.
Manual is generally used only during system startup or during a major
upset condition when the operator must take control to stabilize the pro-
cess. The controller shown in Figure 2-3 has both a horizontal bar display
and a digital indicator to provide the operator with the value of the output
Figure 2-2. Functional block diagram of feedback loop
Control
Valve
Manipulated
Variable
Controller Case
Transmitter
Process
Sensor
Controlled
Variable
Comparator Set Point
Output
Transmission
System
Input
Transmission
System
Controller
Function

signal from the controller. The square indicator marked “RSP” is used to
indicate that the controller is using a remote set point.
Time Elements of a Feedback Loop
The various components of the feedback control loop shown in Figure 2-2
need time to sense an input change and transform this new condition into
an output change. The time of response of the control loop is the combina-
tion of the responses of the sensor, the transmitter, the controller, the final
control element, and the process.
An important objective in control system design is to correctly match the
time response of the control system to that of the process. To reach this
objective, it is necessary to understand the concept of time delays or “lags”
in process control systems.
Time Lags
In process control, the term lag means any relationship in which some
result happens after some cause. In a feedback control loop, lags act in
series, the output of one being the input to another. For example, the lags
Figure 2-3. Typical electronic controller
A
M
OUT
45.1
FIC100
REACTOR FLOW
PV SP
RSP

around a simple temperature control loop would be the output of the elec-
tric controller to the input to a valve lag. The output of the valve lag is the
input to a process heat lag. The output of process heat lag is the input to
the measurement sensor lag. We will start our discussion of time response
and time lag with sensor time response.
Sensor Time Response
In process sensors, the output lags behind the input process value that is
being measured. Sensor output changes smoothly from the moment a
change in measurement value occurs, even if the disturbance is sudden
and discontinuous. It is interesting to note that the nature of the sensor
time-response curve is the same for virtually all sensors, even though the
sensors measure different physical variables.
A typical response curve for a process sensor is shown in Figure 2-4,
where the input has been changed suddenly at time equal to zero.
This curve is described by the following equation for the output measure-
ment m(t) as a function of time:
m(t) = mi + (mf – mi)(1 – e–t/τ
) (2-1)
where
mi = the initial sensor output measurement
mf = the final sensor output value
τ = the sensor time constant
Note that the sensor output is in error during the transition time of the
output value from mi to mf. The actual process variable was changed
instantaneously to a new value at t = 0. Equation 2-1 relates initial sensor
Figure 2-4. Exponential time response of a sensor
Time (t)
mf
mi
Sensor Output (m)
Change
t = 0

output, final sensor output, and the time constant that is a characteristic of
the sensor. The significance of the time constant τ can be found by looking
at the equation for the case where the initial sensor output is zero. In this
special case, the sensor output value is as follows:
m(t) = mf(1 – e–t/τ
) (2-2)
If we wish to find the value of the output exactly τ seconds after a sudden
change occurs, then
m(τ) = mf(1 – e–1
) (2-3)
m(τ) = 0.632mf (2-4)
Thus, we see that one time constant (1τ) represents the time at which the
output value has changed by 63.2 percent of the total change. If we solve
Equation 2-2 for time equal to 5τ, or five time constants, we find that
m(5τ) = 0.993mf (2-5)
This means that the sensor reaches 99.3 percent of its final value after five
time constants.
The following example illustrates a typical sensor response application.
First-order Lag
The first-order lag is the most common type of time element encountered
in process control. To study it, it is useful to look at the response curves
when the system is subjected to a step input, as shown in Figure 2-5. The
advantage of using a step input as a forcing function is that the input is at
steady state before the change and then is instantaneously switched to a
new value. When the output curve (y) is studied, the transition of the sys-
tem can be observed as it passes from one steady state to a new one. The
output or response to the step input applied at time zero (to) is not a step
output but an output that lags behind the input and gradually tries to
reach some final value.
The equation for the system shown in Figure 2-5 is as follows:
(2-6)
where
y(t) = the output y as a function of time
x(t) = the input x as a function of time
)
(
)
(
)
(
dt
dv
t
Kx
t
y
dt
t
dy c
=
+
τ

K = a constant
τ = the system time constant
The system response is called a “first-order lag” because the output lags
behind the input, and the differential equation for the system shown in
Figure 2-5 is a linear first-order differential equation.
EXAMPLE 2-1
Problem: A sensor measures temperature linearly with a transfer function of
30 mv/°C and has a one-second time constant. Find the sensor output two
seconds after the input changes rapidly from 25°C to 30°C. Also find the
process temperature.
Solution: First, find the initial and final values of the sensor output:
mi = (30 mv/°C) (25°C)
mi = 750 mv
mf = (30 mv/°C)(30°C)
mf = 900 mv
Then, use Equation 2-1 to solve for the sensor output at t = 2 s. Note that
e = 2.718.
m(t) = mi + (mf – mi)(1 – e–t/τ
)
m(2) = 750 mv + (900 - 750) mv(1 – e–2
)
m(2) = 879.7 mv
This corresponds to a process temperature at t = 2 s of
T(2) = (879.7 mv)/(30 mv/°C)
T(2) = 29.32°C
Figure 2-5. Response of a system to step
Step Input
System
y
to
to
x
Output

Differential equations are difficult to understand in some cases. If the sys-
tem as a whole contains several components with their own differential
equations, it is very difficult to understand or solve the entire system.
The French mathematician Pierre-Simon Laplace developed a method to
transform differential equations into algebraic equations so as to simplify
the calculations for systems governed by differential equations. We will
avoid most of the rigorous math of the Laplace transform method and
simply give the steps required to transform a normal differential equation
into an algebraic equation:
1. Replace any derivative symbol, d/dt, in the differential equation
with the transform symbol s.
2. Replace any integral symbol, ...dt, with the symbol 1/s.
3. Replace the lowercase letters that represent variables with their
corresponding uppercase letter in the transformed equation.
We can use the Laplace transform method to convert differential Equation
2-6 for our system into an algebraic equation. Since the system equation
contains only a single derivative and no integral, it can be transformed
using steps 1 and 3. When we transform the equation, it becomes
τsY(s) + Y(s) = KX(s) (2-7)
The transfer function for our system is defined as follows:
(2-7a)
Thus, solving Equation 2-7 results in the following equation:
(2-7b)
This is the form of a first-order lag system. First-order lag systems in pro-
cess applications are characterized by their capacity to store matter or
energy. The dynamic shape of their response to a step input is a function of
their time constant. This system time constant, designated by the Greek
letter τ (tau), is meaningful both in a physical and a mathematical sense.
Physically, it determines the shape of the response of a process or system
to a step input. Mathematically, it predicts, at any instant, the future time
period that is required to obtain 63.2 percent of the change remaining. The
response curve in Figure 2-6 illustrates the concept of a time constant by
showing the response of a simple first-order system to a step input. In this
system, the output is always decreasing with time, that is, the rate of
∫
)
(
)
(
s
X
s
Y
Input
Output
=
1
)
(
)
(
+
=
s
K
s
X
s
Y
τ

Exploring the Variety of Random
Documents with Different Content

most assuredly contribute to the advancement and elevation of the
vocal art, if gifted children, as it often happened in former times,
were early instructed in singing with the requisite care and skill.
Thus, educated for their art, and giving to it their best powers, they
would be able to satisfy far higher demands and attain to quite
another and higher artistic perfection than we are wont now-a-days
to find anywhere among our vocal artists. Such children would then,
at the age at which at present instruction 180 in singing begins, have
already mastered all technical difficulties and be able to apply
themselves chiefly to the æsthetic cultivation of their art. With young
girls especially, whose vocal organs do not change so much as those
of boys, the earliest possible beginning of instruction would be in the
highest degree advantageous. It is owing only to the unnatural,
overstrained method of studying the art of singing now prevalent
that a principle recognized and applied in the learning of all other
arts, and even in all the other branches of music, has universal
prejudice against it.
CONCLUSION
An artist can be formed only by his own intelligence and practice,
under the direct guidance of a master. But here, more than in any
other art, the constant watchfulness of a teacher is a necessity. For,
as one gets only an imperfect idea of his own personal appearance
from a mirror, so the singer and dramatic artist can form but a
partial judgment of his own performances. They are too subjective,
and cannot be viewed as an external whole, like the works of the
painter and sculptor. It is, moreover, as has 181 already been

remarked, simply impossible to obtain even a partial knowledge of
any art from books alone, even if we were able to describe with
precision the fine, delicate differences of tones, colors and forms.
These pages, therefore, make no claim whatever to be regarded
as a manual of singing. They aim only to communicate and extend a
knowledge of the latest discoveries and advances in the domain of
vocal art, and to protest against and correct prevailing prejudices
and errors in regard to this art, as well as to engage the attention of
those to whose care the culture of the voice is entrusted.
18 The friends of this style of music (programme music so called) appeal to
the authority of Beethoven, who, it is claimed, opened the way for it
when he introduced into his Pastoral Symphony interlineations which
should suggest the right sentiment to the hearer. But, although
Beethoven allowed himself to approach the uttermost limits in this
direction, he never overstepped them. It was only in his Pastoral
Symphony that he introduced these interlineations, and they do not
entirely contradict the peculiar character of the music, as so many of our
modern programmes do.
Programme
To Beethoven’s Pastoral Symphony, December 22, 1808.
I. Agreeable sensations upon visiting the country.
II. Scene at a brook’s side.
III. Merry gathering of country people.
IV. Thunder and storm.
V. Happy and grateful emotions after the storm.
More emotional than descriptive.
Expression rather than representation of feeling.

Programme
To a Prize Symphony, by Joachim Raff, performed in Vienna, 1863.
I. D major. Allegro.
Portrait of the German character,—its capability of elevation,
proneness to Reflection, Gentleness and Valor, as contrasts
that blend with and permeate one another in manifold ways—
overpowering proneness to meditation.
II. D minor. Allegro molto vivace.
In the open air, in the German grove, with the sound of horns,
Away to the fields, with the songs of the people.
III. D major. Larghetto.
Gathering round the domestic hearth, transfigured by love and
the Muses.
IV. G minor. Allegro-dramatico.
Ineffectual struggle to establish the unity of the fatherland.
V. D minor. Lament. D major. Allegro trionfale.
Opening of a new and elevated era.
Return to text
19 Although our recitative is formed after the recitative of the ancient
drama, yet the latter, according to all accounts, appears to have been
very different from our opera recitative, and to have had greater
resemblance to the monotonous recitation of the Romish Liturgy, which
seems to be a relic of ancient art.
Return to text

183 APPENDIX
185 STRUCTURE OF THE VOCAL ORGANS
The larynx is a sound-giving organ belonging to that class of wind
instruments called reed instruments, although it differs in various
respects from all artificial arrangements of the kind. The sound or
tone-generating apparatus of the larynx consists of tense, elastic
membranes, the so-called chordæ vocales, which are enclosed in a
sounding case composed of movable cartilaginous plates, and may
be stretched by a certain apparatus of muscles in very different and
exactly measurable degrees. They are made to vibrate audibly by a
current of air impelled with various degrees of force and at will by
the lungs in expiration through the narrow chink (glottis) formed by
the fine edges of the chords. Thus the lungs correspond to the
bellows of the organ; the trachea, at the top of which the vocal
instrument is placed, answers to the conduit (Windrohr), and the
cavity of the throat in front of the instrument with its two avenues,
the mouth and the nostrils, to the resonance pipe (Ansatzrohr).

186 THE LUNGS
The lungs are two cellular, sponge-like elastic organs, largely
made up of little cavities of conical shape, which, in the regular
alternations of two opposite respiratory movements of air, are at one
time expanded, and then again compressed. The two lungs are not
of equal size; the right lung is one-tenth larger in volume than the
left.
THE TRACHEA, OR WINDPIPE,
Through which the air of the lungs enters and passes out, consists
of from sixteen to twenty-six cartilaginous rings, posteriorly
incomplete, lying horizontally one above the other.
These rings are connected by a membrane covering them
externally and internally. As they enter the cavity of the chest, they
divide into two branches, likewise composed of rings, one entering
the right, the other the left lung. Before they join the lungs they
divide again into several smaller branches, which again subdivide
fork-like in the lungs, and terminate in numberless little grape-like
clusters of hollow vesicles. The diameter of the trachea in adults is
from one-half to three-fourths of an inch when at rest.
THE LARYNX
The larynx may be regarded as the funnel-shaped termination of
the trachea. It enlarges upward and is composed of various
cartilages more or less mobile, connected by ligaments and moved
by muscles. The exterior of the larynx is formed by the

I. 187 Thyroid cartilage.
II. Cricoid cartilage.
The cartilages in the interior are:
I. The Arytenoid cartilages.
II. Cartilages of Wrisberg.
III. Cartilages of Santorini.
IV. Cuneiform cartilages.
To the cartilages of the larynx must be further added the
Epiglottis, with the little cartilage at the centre of its inner side.
1. The thyroid cartilage is the largest cartilage of the larynx, and
consists of two four-cornered cartilaginous plates held together in
front and diverging behind; the anterior borders are convex, and
consequently where the two plates meet in front they form an upper
and a lower notch or slit. The posterior angles of this cartilage
extend into the so-called horns of the thyroid cartilage. At the upper
horns are ligaments attached, which form the connection between
the hyoid bone and the larynx, while the lower horns serve to join
the thyroid to the cricoid cartilage. In females and boys the angle
formed by the two plates of the thyroid cartilage is obtuse. In the
male sex at a certain period the larynx changes its shape, and the
plates of the thyroid cartilage then form an acute angle, which is
visible on the outside of the throat, and is popularly known as the
Adam’s apple. At this time the diameter of the male larynx becomes
a third larger than that of the female larynx, and in consequence the
voice is lower, and its different registers are more enlarged in
compass.
188 2. The cricoid cartilage resembles in shape a seal ring; its
broader side is situated posteriorly between the lower horns of the

thyroid cartilage, and it is connected by its lower edges immediately
with the upper edge of the first ring of the trachea. From its side at
the back part project two rounded surfaces, which give attachment
to the arytenoid cartilages.
3. The arytenoid cartilages are two small but very mobile bodies in
the form of three-cornered pyramids. The base of the pyramid rests
upon the before-mentioned rounded surface at the back of the
upper border of the cricoid cartilage; one of its sides turns to the
front, the two others to the back and outwards. The surfaces
between the anterior and postero-interior corners are accordingly
turned towards one another. The surface posteriorly is concave, and
affords space for a part of the arytenoid muscle; the inner surface is
smooth, and forms, during quiet breathing, a part of the lateral wall
of the larynx; the anterior surface is rough and irregular, and to it
adhere the vocal chords, the thyro-arytenoid muscle, the lateral and
posterior crico-arytenoid muscles, and upon these the bases of the
cuneiform cartilages. The arytenoid cartilages are lengthened at
their summits by two little pear-shaped elevations, the cartilages of
Santorini (called apophyses in Garcia’s observations), which are
connected with them by ligamentous fibres, and extend with them
some distance into the larynx.
4. The cartilages of Wrisberg are described by Hyrtl as slight
elevations upon the front or anterior edge of 189 the arytenoid
cartilages, inclining towards the interior, and, like all parts of the
larynx, covered by the mucous membrane.
5. The cuneiform cartilages (as Wilson names them) are two long,
slender cartilaginous laminæ which become somewhat broader at
both ends. These cartilages, with their base, rest in the middle of

the anterior surface of the arytenoid cartilages, and reach to the
middle of the vocal chords, by which they are enveloped. The action
of these cartilages renders possible the production of the head
tones, but they are not found in every larynx. The fact that they are
oftener found in the female larynx than in that of the male, and that
the male larynx is mostly used in scientific investigations, as it is
larger and more easily dissected, may be the reason why up to the
present time no mention is made of them either in German or
French manuals. They are sometimes referred to as cuneiform
cartilages, or confounded with the cartilages of Wrisberg, probably
because it seemed unaccountable that these important bodies
should so long have escaped the attention of anatomists.
From the anterior surface of the arytenoid cartilages, extending
towards the centre of the inner wall of the thyroid cartilage, running
diagonally through the cavity of the larynx, are stretched the two
pairs of chords already more than once mentioned—the vocal
chords, consisting of folds of the mucous membrane which
envelopes the whole larynx. The two lower of these chords, the
vocal chords strictly so called, into which the cuneiform cartilages
project and through which the interior thyro-arytenoid 190 muscles
run, have their points of attachment at the arytenoid cartilages,
somewhat lower than the upper pair. Each of these parallel pairs of
chords form between their lips a slit running antero-posteriorly. The
slit of the upper pair is opened in the shape of an ellipse; that of the
lower pair, the glottis, is very narrow. As the upper chords have their
point of attachment posteriorly and higher, they form with the lower
chords two lateral cavities, the ventricles.

The two pairs of chords, therefore, are the free interior edges of
the membrane, covering the whole larynx and extending into it to
the right and the left. Only the lower vocal chords serve directly for
the generation of tones. More or less stretched and presenting
resistance to the air forcibly expired from the lungs through the
trachea, they are thus made to vibrate. The upper or false vocal
chords do not co-operate with them to generate tone, but like all the
remaining parts of the mouth and throat belong to the resonance
apparatus of the voice, to which also appertains the back part of the
mouth, the pharynx, over the œsophagus, the throat, or gullet. This
is separated from the anterior cavity of the mouth by the palate,
which is a curtain formed by the mucous membranes of the cavity of
the mouth, and the centre of which forms the pendent uvula.
Above the œsophagus, immediately over the palate, lie close
together, and separated only by a very thin osseous partition, the
two posterior nasal orifices. These serve as passages for the air
during inspiration and expiration; 191 they are likewise considered as
belonging to the resonance apparatus.
Upon both sides of the cavity of the mouth, between the two
wings of the palate, lie the tonsils, two glandular bodies, which
separate the sides of the cavity of the mouth from the pharynx. The
anterior cavity of the mouth, which is separated from the nasal
cavities by the palate, requires no description, as every one can
acquaint himself with its structure in his own person and in others.
Upon its formation, as well as upon the position of its different parts
and upon the character of those parts of the larynx and of the cavity
of the mouth which have been described as the resonance

apparatus, the difference in the fulness and timbre of tones
depends.
The epiglottis is fixed at the anterior portion of the larynx, at the
root of the tongue, within the angle formed by the two surfaces of
the thyroid cartilage. It is a very elastic fibro-cartilage, freely moving
in a posterior direction. Its color is yellowish and its general form
that of a spoon; its upper surface is covered with a multitude of little
mucous glands set in shallow cavities. In the downward passage of
food the epiglottis covers the upper orifice of the larynx like a valve,
over which the food passes into the œsophagus or gullet, without
being able to enter the larynx and the trachea. In the centre of its
interior side there is a little rounded cartilage, movable in every
direction, which has as yet no name. Czermak mentions it first in his
observations with the laryngoscope. In the male larynx, after the
voice has altered, the cartilages become 192 more or less ossified and
gradually harden with increasing age. The cartilages of the female
larynx, with rare exceptions, usually continue with little or no
change. The muscles, by which the movements of the larynx are
effected, are:
I. The posterior crico-arytenoid.
II. The lateral crico-arytenoid.
III. The crico-arytenoid.
IV. The thyro-arytenoid.
V. The arytenoid.
VI. The internal thyro-arytenoid.
In late works upon laryngoscopy the different muscles of the
larynx are variously designated and divided. Bataille terms the first
three of the above-named muscles the exterior muscles of the
larynx; the three others he comprehends under the name of thyro-

arytenoid or vocal muscle, which divides into three slips in the
interior of the larynx. This, however, as well as the description of the
character and action of the different muscles, belongs to the
department of science. What I have already stated seems to me to
be sufficient for an understanding of the action of these organs in
the production of sound in the different registers. The reader is
referred to any good manual of anatomy for a full description of the
muscles, ligaments, nerves, vessels and membranes.
Return to page 41, page 60, page 61
THE END.

Transcriber’s Note.
The following amendments have been made to the original text:
Page 24: “and were the forerunners of the opera and oratorio”
for “and were the fore-/runners of the opera and oratorio”
Page 26: “But in that brilliant springtime of vocal art”
for “But in that brilliant spring-/time of vocal art”
Page 30: “i. e., the old singing masters taught”
No comma in original (added for consistency).
Page 42: “by their anterior apophyses, without leaving any
space”
Comma missing from original.
Page 48: “vol. x. 4th Series, pp. 218–221, 1855.”
for “vol. x. 4th Series, p. 218–221, 1855.”
Page 66: “i. e., those tones where a different action”
No comma in original (added for consistency).
Page 79: “the tones of the head register.”
Period missing from original.
Page 87: “the reed of the mouthpiece.”
for “the reed of the mouth-/piece.”
Page 108: “the air coming from the lungs through the trachea,”
for “the air coming from the lungs through the treachea,”

“then the high inharmonic over-tones are prominent,”
for “then the high inharmonic overtones are prominent,”
Page 132:
for (compare fourth note)
Footnote 18: “struggle to establish the unity of the fatherland.”
for “struggle to establish the unity of the father-/land.”
Textual representation of the diagram on
page 68
IN THE MALE VOICE
TENOR
VOICE
First series of the chest
register.
Second
series.
First series of
the falsetto.
G A B c d e f g a b c1
d1
e1
f1
g1
a1
, &c.
C D E F G A B c d e f g a b c1
d1
e1
f1
First series of the chest register.
Second
series. BASS VOICE

IN THE FEMALE VOICE
First series
of the chest
register.
Second
series of
the
chest
register.
First series
of the
falsetto
register.
Second
series of
the
falsetto
register.
Head register.
e f g a b c1
d1
e1
f1
g1
a1
b1
c2
d2
e2
f2
g2
a2
b2
c3
d3
e3
f3

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