Standard

ISA37.16.01

A Guide for the DynamicCalibration of PressureTransducers

Approved Date

ISA37.16.012002A Guide for the Dynamic Calibration of Pressure Transducers

ISBN:

Copyright 2002 by ISA The Instrumentation, Systems, and Automation Society. All rights reserved.Not for resale. Printed in the United States of America. No part of this publication 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.

ISA67 Alexander DriveP. O. Box 12277Research Triangle Park, North Carolina 27709

- 3 - ISA-37.16.01-2002

Preface

This preface, as well as all footnotes and annexes, is included for information purposes and is not part ofISA-37.16.01-2002.

This document has been prepared as part of the service of ISA the Instrumentation, Systems, andAutomation Society toward a goal of uniformity in the field of instrumentation. To be of real value, thisdocument should not be static but should be subject to periodic review. Toward this end, the Societywelcomes all comments and criticisms and asks that they be addressed to the Secretary, Standards andPractices Board; ISA; 67 Alexander Drive; P. O. Box 12277; Research Triangle Park, NC 27709;Telephone (919) 549-8411; Fax (919) 549-8288; E-mail: standards@isa.org.

The ISA Standards and Practices Department is aware of the growing need for attention to the metricsystem of units in general, and the International System of Units (SI) in particular, in the preparation ofinstrumentation standards. The Department is further aware of the benefits to USA users of ISAstandards of incorporating suitable references to the SI (and the metric system) in their business andprofessional dealings with other countries. Toward this end, this Department will endeavor to introduceSI-acceptable metric units in all new and revised standards, recommended practices, and technicalreports to the greatest extent possible. Standard for Use of the International System of Units (SI): TheModern Metric System, published by the American Society for Testing & Materials as IEEE/ASTM SI 10-97, and future revisions, will be the reference guide for definitions, symbols, abbreviations, andconversion factors.

It is the policy of ISA to encourage and welcome the participation of all concerned individuals andinterests in the development of ISA standards, recommended practices, and technical reports.Participation in the ISA standards-making process by an individual in no way constitutes endorsement bythe employer of that individual, of ISA, or of any of the standards, recommended practices, and technicalreports that ISA develops.

CAUTION ISA ADHERES TO THE POLICY OF THE AMERICAN NATIONAL STANDARDSINSTITUTE WITH REGARD TO PATENTS. IF ISA IS INFORMED OF AN EXISTING PATENT THAT ISREQUIRED FOR USE OF THE DOCUMENT, IT WILL REQUIRE THE OWNER OF THE PATENT TOEITHER GRANT A ROYALTY-FREE LICENSE FOR USE OF THE PATENT BY USERS COMPLYINGWITH THE DOCUMENT OR A LICENSE ON REASONABLE TERMS AND CONDITIONS THAT AREFREE FROM UNFAIR DISCRIMINATION.

EVEN IF ISA IS UNAWARE OF ANY PATENT COVERING THIS DOCUMENT, THE USER ISCAUTIONED THAT IMPLEMENTATION OF THE DOCUMENT MAY REQUIRE USE OF TECHNIQUES,PROCESSES, OR MATERIALS COVERED BY PATENT RIGHTS. ISA TAKES NO POSITION ON THEEXISTENCE OR VALIDITY OF ANY PATENT RIGHTS THAT MAY BE INVOLVED IN IMPLEMENTINGTHE DOCUMENT. ISA IS NOT RESPONSIBLE FOR IDENTIFYING ALL PATENTS THAT MAYREQUIRE A LICENSE BEFORE IMPLEMENTATION OF THE DOCUMENT OR FOR INVESTIGATINGTHE VALIDITY OR SCOPE OF ANY PATENTS BROUGHT TO ITS ATTENTION. THE USER SHOULDCAREFULLY INVESTIGATE RELEVANT PATENTS BEFORE USING THE DOCUMENT FOR THEUSERS INTENDED APPLICATION.

HOWEVER, ISA ASKS THAT ANYONE REVIEWING THIS DOCUMENT WHO IS AWARE OF ANYPATENTS THAT MAY IMPACT IMPLEMENTATION OF THE DOCUMENT NOTIFY THE ISASTANDARDS AND PRACTICES DEPARTMENT OF THE PATENT AND ITS OWNER.

ADDITIONALLY, THE USE OF THIS DOCUMENT MAY INVOLVE HAZARDOUS MATERIALS,OPERATIONS OR EQUIPMENT. THE DOCUMENT CANNOT ANTICIPATE ALL POSSIBLEAPPLICATIONS OR ADDRESS ALL POSSIBLE SAFETY ISSUES ASSOCIATED WITH USE IN

ISA-37.16.01-2002 - 4 -

HAZARDOUS CONDITIONS. THE USER OF THIS DOCUMENT MUST EXERCISE SOUNDPROFESSIONAL JUDGMENT CONCERNING ITS USE AND APPLICABILITY UNDER THE USERSPARTICULAR CIRCUMSTANCES. THE USER MUST ALSO CONSIDER THE APPLICABILITY OFANY GOVERNMENTAL REGULATORY LIMITATIONS AND ESTABLISHED SAFETY AND HEALTHPRACTICES BEFORE IMPLEMENTING THIS DOCUMENT.

THE USER OF THIS DOCUMENT SHOULD BE AWARE THAT THIS DOCUMENT MAY BE IMPACTEDBY ELECTRONIC SECURITY ISSUES. THE COMMITTEE HAS NOT YET ADDRESSED THEPOTENTIAL ISSUES IN THIS VERSION.

The following people served as members of ISA Subcommittee SP37.16:

NAME COMPANY

L. Whitby, Chairman DeVry Institute of TechnologyJ. Weiss, Managing Director KEMA ConsultingJ. Branom Branom Instruments CompanyH. Estrada BF Goodrich-Advanced Micro MachinesM. Montreuil National Research Council CanadaR. Rhen PCB PiezotronicsR. Staus Pennsylvania State UniversityT. Vondenbenken Kulite SemiconductorP. Walter TCU Endevco Corporation______

* One vote per company.

The following people served as members of ISA SP37 Committee:

NAME COMPANY

E. Icayan, Chairman ACES Inc.J. Weiss, Managing Director KEMA ConsultingT. Anderson SpaceAge Control Inc.C. Flagg ACSJ. Hendrie Lucas Control Systems ProductsA. Mobley 3M CompanyM. Montreuil National Research Council CanadaH. Norton Jet Propulsion LabR. Staus Pennsylvania State UniversityP. Walter TCU Endevco CorporationL. Whitby DeVry Institute of TechnologyJ. Wilson The Dynamic Consultant LLCW. Zubon Bently Nevada Corporation______

* One vote per company.

This draft standard was approved for publication by the ISA Standards and Practices Board on_________________.

NAME COMPANY

M. Zielinski, Chair Emerson Process ManagementD. Bishop David N Bishop, ConsultantD. Bouchard PapricanM. Cohen Consultant

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M. Coppler Ametek, Inc.B. Dumortier Schneider ElectricW. Holland Southern CompanyE. Icayan ACES IncA. Iverson Ivy OptiksR. Jones Dow Chemical CompanyV. Maggioli Feltronics CorporationT. McAvinew ForeRunner CorporationA. McCauley, Jr. Chagrin Valley Controls, Inc.G. McFarland Westinghouse Process Control Inc.R. Reimer Rockwell AutomationJ. Rennie Factory Mutual Research CorporationH. Sasajima Yamatake CorporationI. Verhappen Syncrude Canada Ltd.R. Webb POWER EngineersW. Weidman Parsons Energy & Chemicals GroupJ. Weiss KEMA ConsultingM. Widmeyer Stanford Linear Accelerator CenterC. Williams Eastman Kodak CompanyG. Wood Graeme Wood Consulting

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CONTENTS

Acknowledgments from the Current ISA SP37.16 Subcommittee................................................................ 9

Introduction ................................................................................................................................................. 11

1 Scope................................................................................................................................................... 13

2 Purpose................................................................................................................................................ 13

3 Table of Symbols ................................................................................................................................. 13

4 Transducer properties.......................................................................................................................... 13

4.1 Underdamped second-order systems .......................................................................................... 14

4.2 General transducer properties...................................................................................................... 19

4.3 Properties in the frequency domain.............................................................................................. 20

4.4 Properties in the time domain....................................................................................................... 20

5 Dynamic pressure generators ............................................................................................................. 22

5.1 Shock tube.................................................................................................................................... 22

5.2 Shockless pressure-step generators............................................................................................ 27

5.3 Pulse generators .......................................................................................................................... 28

5.4 Periodic pressure function generators (sinusoidal pressure generators)..................................... 29

6 Measurement of transducer properties................................................................................................ 33

6.1 Sensitivity ..................................................................................................................................... 33

6.2 Amplitude response...................................................................................................................... 33

6.3 Phase response............................................................................................................................ 35

6.4 Resonant frequency ..................................................................................................................... 36

6.5 Ringing frequency......................................................................................................................... 36

6.6 Damping ratio ............................................................................................................................... 36

6.7 Rise time....................................................................................................................................... 37

6.8 Overshoot ..................................................................................................................................... 38

7 Transducer interfaces.......................................................................................................................... 38

ISA-37.16.01-2002 - 8 -

7.1 Mounting, strain effects ................................................................................................................ 38

7.2 Cavities and passages ................................................................................................................. 39

7.3 Temperature effects ..................................................................................................................... 40

7.4 Acceleration effects ...................................................................................................................... 42

8 Electronic considerations..................................................................................................................... 43

8.1 Noise............................................................................................................................................. 43

8.2 Cabling ......................................................................................................................................... 44

8.3 Voltage amplifier........................................................................................................................... 45

8.4 Charge amplifier ........................................................................................................................... 46

9 Data acquisition and analysis .............................................................................................................. 47

9.1 Digital oscilloscope or recorder .................................................................................................... 47

9.2 Data analysis ................................................................................................................................ 48

10 Reporting test results ....................................................................................................................... 48

10.1 Test conditions ............................................................................................................................. 48

10.2 Results and discussion................................................................................................................. 49

11 References....................................................................................................................................... 49

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Acknowledgments from the Current ISA SP37.16 Subcommittee

This document has a long history. Originally published by the American Society of Mechanical Engineersas A Guide for the Dynamic Calibration of Pressure Transducers, this document been an ANSI Standard since 1972. The theory, including the physics and mathematics of dynamic calibration, is timeless, but thefurther development of technology has caused changes in some of the methods described in the originaldocument.

In 1996, the ASME handed the document over to the ISA. ISAs SP37.16 Subcommittee on PressureTransducers began editing the document in 1997, in order to update the methods and references whileleaving the timeless aspects alone. The result is ISA- 37.16.01- 2002, A Guide for the Dynamic Calibration of Pressure Transducers.

The following SP37.16 Subcommittee members are recognized because of the major involvement theyhad with the current revision of this document, and more importantly because of the role they might serveas reference sources in any future updates: Patrick Walter, Jim Lally, and Bob Goodemote. However, thisdoes not lessen the contributions of the entire committee in this effort.

As stated in the original abstract: "While not intended as a step-by-step procedure, this document doescontain specific examples and suggested methods for the determination of items of interest in thecalibration of dynamic pressure transducers."

It is quite likely that in a very short period of time, another subcommittee will edit this document to reflectcurrent technologies and techniques available.

Lawrence WhitbySP 37.16 Subcommittee Chair

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Introduction

The state of art of dynamic pressure calibration and pressure sensor technology has significantlyadvanced since the original publication of this document in 1972 (Reference 56). The ASME standarddocuments early attempts to develop dynamic pressure calibration methods, some of which neverevolved further into successful technology. Most of the calibration devices described in this documentwere uniquely engineered at individual laboratories to meet their specific measurement needs. However,today, some of these devices have evolved into commercially available products.

The need to measure "nonsteady" dynamic pressure became very important after WW-II during the rapiddevelopment of jet aircraft and aerospace technology. Investigations of turbulence associated withlaunch, shock waves upon re-entry, sonic boom, rocket combustion stability, air blast (References 67,68), and the dynamics involved with weapons testing were significant measurement challenges.Investigations in these and other areas have necessitated faithful measurement of pressure variations atfrequencies from near zero to the neighborhood of 106 Hertz (Hz). The degree of accuracy with whichthese measurements must be made varies widely throughout the technical community, as does the usemade of information derived from such measurements. Often there are other complicating factors, suchas severe environmental effects, which must be considered, if meaningful information is to be obtained.When considering the measurement problem, the investigator must first determine the dynamiccharacteristics of the pressure transducer. It is toward the satisfaction of this basic requirement that thisdocument is directed.

Dynamic pressure calibration at the time this document was originally authored was difficult because ofthe limitation of dynamic pressure calibration sources available. Dynamic calibrators were simply notcommercially available. Since then, substantial improvement has been made in the state-of-the-art ofboth dynamic pressure calibrators and high-frequency pressure transducers to meet many currentmeasurement requirements for amplitude, frequency, and accuracy. Most of the dynamic calibratorsavailable today incorporating fast-acting valves yield dynamic pressure amplitudes that are independentlyestablished. Others use a pressure transducer as a "transfer" standard that the transducer beingcalibrated is compared against (References 56, 57).

Although the users requirement for information concerning a transducers response characteristics hasbeen as varied as the test methods used to obtain the data, current commercial calibrators and digitaldata acquisition systems have helped to obtain more accurate information. Unfortunately there have beenmany instances where worthwhile data have gone unused because of the manner in which they werepresented. Test reports lacking adequately defined terms, test conditions, or other supporting informationconvey little more than misunderstanding to the reader.

The intent of this document is to provide documentation for current techniques and to identify possiblepitfalls associated with the dynamic calibration of pressure transducers. The results of providing such adocument to the technical community will be a better understanding of the basic problems as well asmore effective communication between workers in the field.

This document is not a step-by-step procedure that can be followed without fail to the absolute truth inpressure measurements. Neither is it an attempt to discuss in detail all of the factors that affect theaccuracy of pressure measurements, e.g., environmental effects, signal transmission, or recordingtechniques. References to applicable documents concerning such problems are contained herein, andReference 74 deals specifically with the measurement/data acquisition/data utilization process. Thisdocument concentrates on the factors that directly affect dynamic response, such as adapters andmechanical attachments physically a part of, or relatively inseparable from the transducer, and electronicequipment that, in practical use, is required for the operation of the transducer. The description ofequipment and techniques appearing in this document will be limited to their use as directly related todynamic pressure calibration.

ISA-37.16.01-2002 - 12 -

The clauses of this document are divided into three groups. The first, consisting of Clauses 4, 5, and 6,discusses the significant transducer properties, dynamic pressure sources available, and the use ofsources to determine the desired transducer properties. The second group, consisting of Clauses 7 and8, deals with the problems of transducer installation and the immediate electronic signal conditioningnecessary to obtain a satisfactory output signal. The final group, consisting of Clauses 9 and 10, indicatesdata-recording methods and recommends procedures for reporting test results.

Although this document focuses primarily on pressure levels above acoustic, it is worth noting that inchprecision condenser microphones, in compliance with ISA 1094-1-4, also have been successfully used fordynamic pressure measurements on jet aircraft, rocket engines, and other aerospace applications. Fordynamic calibration, the open-circuit sensitivity and the frequency response are normally obtained by aprecision acoustical calibration system (Reference 51) using a pistohphone (Reference 52) for the open-circuit sensitivity, and the electrostatic-actuator method for the frequency response. An acousticalcalibrator (Reference 53) meeting the requirements of IEC 942 (1988) Class 1 may also be used tocalibrate the open-circuit sensitivity.

This document is inconsistent in the use of the word calibration in reference to dynamic testing ofpressure transducers. It should be understood that calibration as used in this document and others(Reference 1) means a test during which known values of measurand are applied to a transducer, andcorresponding output readings are recorded. The degree of accuracy associated with these dynamic testsis generally lower, and the manner in which the results are used is generally less rigorous than in theconventional and more easily controllable field of static pressure calibration.

In preparing this Guide in 1972, the original ANSI B88 Subcommittee on Pressure had considered thevarious testing and reporting techniques before recommending specific practices. This present documentrepresents the first step in the accomplishment of the Subcommittees assignment, which was to developor approve standards for the dynamic calibration of pressure transducers in order to improve the quality ofdynamic calibrations.

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1 Scope

This standard covers dynamic pressure transducers, which are, primarily those used in measurements.

2 Purpose

This standard establishes guidelines for the preferred techniques and practices in the calibration ofdynamic pressure transducers.

3 Table of Symbols

a gas speed of sound RC time constant of R-C circuiteA outlet orifice area st settling time

eA maximum exit area stD shock-wave transit timeiA inlet orifice area rt rise time, transducer

rA amplification factor T temperaturec damping v chamber volumed diameter V voltage or volumef frequency VD peak incremental voltagek spring constant V peak voltage for any cycleK steady-state sensitivity V average voltagel piston position pV peak-output voltageL length of cylindrical passage sV shock-wave velocitym mass modulation factor

sM shock wave Mach number g ratio of specific heats of constantN number of oscillations pressure and volume

)s(IN)s(OUT transfer function h constant

p pressurez

damping ratiosp stagnation pressure of supply gas l wavelengthap absolute pressure t rise time, inputPD pressure change w frequency in radians per secondop equilibrium pressure dw ringing frequency

p average chamber pressure ow natural frequencyrw resonant frequency

4 Transducer properties

The transducer properties or characteristics of interest to a user will depend to a large extent on theapplication involved. This clause defines and discusses some of the properties most often required.These properties sometimes can be described in terms of the transient response of the device to a stepinput, or in terms of its steady-state response to sine-wave excitation, or both.

In defining transducer properties related to dynamic response, the transducer's transfer function providesvaluable information. The transfer function is the ratio of output to input (expressed in the frequencydomain), and forms the basis for the frequency response parameters. Once the transfer function isknown, the input vs. time for any output can be determined. These topics have been treated by thoseworking in the fields of servomechanisms and network theory, where it is often necessary to describe

ISA-37.16.01-2002 - 14 -

system behavior in both transient and steady-state terms. This approach, using the transfer-functionconcept, has much to offer in the consideration of dynamic calibration of pressure transducers.

Because of the limitations of periodic pressure generators, responses to aperiodic pressure generatorsmust be depended upon to provide much of the needed information on transducers. Measurements offrequency response using sine-wave pressure inputs are easily defined and understood, whereas thenecessity to convert from the time to the frequency domain makes analysis with aperiodic inputs moredifficult.

Description of pressure transducer dynamic properties is usually based on representation of thetransducer as a linear second-order system with a single degree of freedom, e.g., a simple spring masssystem with damping. Some transducers may be found to be more complex than such a simple system,and their analysis is correspondingly more difficult. A detailed analysis of a simple single mass and singlespring system follows.

4.1 Underdamped second-order systems

The typical mass-spring mechanical system, which provides the first resonance of a transducer, isdescribed by a linear second-order differential equation:

(Eq. 4.1)m

)t(fm

kxdtdx

m

c

dtxd2

2=++

where c indicates the damping, k the spring constant, m the mass, and f(t) the forcing function (providedby the pressure generator). The use of the Laplace Transform allows the formation of the transfer functionfrom Equation 4.1.

(Eq. 4.2) 2oo

2

2o

s2sK

)s(IN)s(OUT

w+wz+

w

=

where OUT(s) is the Laplace Transform of the output,IN(s) is the Laplace Transform of the input,K is the steady-state sensitivity,

w o is the natural frequency of the system in radians per second = m

k.

s is the complex variable = j (2p f

) where f is frequency

z is the damping ratio (ratio of actual damping to critical damping)

- 15 - ISA-37.16.01-2002

The natural frequency is the frequency of free (not forced) oscillations of the sensing element of atransducer without damping (c = 0). In practical terms it is the measured frequency at which thetransducer has a 90o phase shift.

The response of an underdamped second-order system is treated in numerous texts. (See References 2,3, 4, and 5.) The amplitude and phase response vs. frequency is shown in Figure 1 and Figure 2, and thetime response to a step input is shown in Figure 3.

The model described by Equation 4.2 assumes that the transducer can respond to static pressures(j w = 0) with sensitivity K. Often transducer systems along with their associated electronics do notrespond to static pressures, and in this case, the model can be modified by incorporating the equivalentof a high-pass RC stage with the transfer function. For example, a piezoelectric transducer with adominant mechanical resonance can be approximated by the transfer function

(Eq. 4.3)RC/1s

s

s2sK

)s(IN)s(OUT

2oo

2

2o

+

w+wz+

w

=

Now when s = 0, the response is zero.

Equations 4.2 and 4.3 are written in terms of the variable s and describe the system behavior in thefrequency domain. When the input function is specified, the inverse Laplace Transform can be used toderive an equation yielding the time domain function. If we apply a pressure step of height A to thetransducer described by Equation 4.2, the time response of the output voltage will be

(Eq. 4.4)

z

z-

+wz-

-=

z-

wz

2

d1

t

2

1tanarctsine

1

11A)t(e 2d

where w d = w o 21 z- .

Equation 4.4 is plotted for various z in Figure 3, and yields the time response of the system to a specifiedinput.

Information on transducer sensors and physical principles can be found in References 2 and 6-12.Information supporting the preceding analysis can be found in Reference 63.

The above model will next be applied to the description of properties of dynamic pressure transducers.

ISA-37.16.01-2002 - 16 -

108654321.00.80.60.50.40.30.20.10.5-40

-30

-20

-10

0

+10

+2010

3

1

0.3

0.1

0.03

0.01

Dec

ibe

ls

Ampl

itude

Fa

ctor

= 0.6V

V = 0.05

= 0.1V

V = 0.15

V = 0.2V = 0.25V = 0.3V = 0.4V = 0.5V = 0.8V = 1.0

0

Figure 1 Amplitude response for an ideal second-order system

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108654321.00.80.60.50.40.30.20.10.05-180

-160

-140

-120

-100

-80

0

-60

-40

-20

P

h

a

s

e

S

h

i

f

t

I

n

D

e

g

r

e

e

s

V = 0.4

= 0.05V = 0.1V = 0.15V = 0.2V = 0.25V = 0.3V

V = 0.5

V = 0.8V = 0.6

V = 1.0

/ 0

Figure 2 Phase response for an ideal second-order system

ISA-37.16.01-2002 - 18 -

V = 1.4

V = 0.8

V = 1.2

V = 0.6

V = 1.0

V = 0.4V = 0.2

0 1 2

2

3 4

4

5 6

6

7 8

8

9 10

10

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Ampl

itude

Angle (In Radians)0t

0t

Figure 3 Response of ideal second-order system to stepinput of unit amplitude

- 19 -

4.2 General transducer properties

There are many general properties of transducers that are described in discussions of static pressuremeasurements. Those properties, which have specific application for dynamic measurement, are asfollows (see also Reference 1).

4.2.1 Sensitivity

The ratio of the change in the transducer output to a change in the value of the measurand. Generally,this is expressed in terms of voltage vs. pressure. The sensitivity is represented by the constant K ofEquations 4.2 and 4.3. If the transducer is accurately represented by Equation 4.2, the sensitivity can beestablished by static measurements. If Equation 4.3 is the model, dynamic measurement of sensitivity isrequired.

4.2.2 Linearity

The closeness of a calibration curve to a specified straight line. Equations 4.2 and 4.3 assume thetransducer output to be linear. Any non-linearity is a deviation from the model equation.

4.2.3 Range

The measurand value, over which a transducer is intended to measure, specified by its upper and lowerlimits.

4.2.4 Creep (drift)

A change in output occurring over a specific time period while the measurand and all environmentalconditions are held constant.

4.2.5 Hysteresis

The maximum difference in output, at any measurand value within the specified range, when the value isapproached first with increasing and then with decreasing measurand.

4.2.6 Proof pressure

The pressure that may be applied to the sensing element of a transducer without changing the transducerperformance beyond specified tolerances.

4.2.7 Repeatability

The ability of a transducer to reproduce output readings when the same measurand value is applied to itconsecutively under the same conditions and in the same direction.

4.2.8 Acceleration-Compensation

An accelerometer element internal to the transducer that reduces its sensitivity to motion.

4.2.9 Thermal sensitivity shift

A change in sensitivity of a pressure transducer as a result of a change in steady-state operatingtemperature, expressed as % / C or F.

ISA-37.16.01-2002 - 20 -

4.2.10 Resolution

The smallest discernible signal from a measurement system. It may also be referred to as "threshold.

4.2.11 Noise

Any unwanted signal in the measurement system other than the desired pressure response.

4.3 Properties in the frequency domain

Properties in the frequency domain are described by the transfer function of Equations 4.2 and 4.3.

4.3.1 Amplitude response

The amplitude of the transfer function versus frequency (often called the frequency response). It can becomputed from Equations 4.2 or 4.3 by substituting jw for s and computing the resultant magnitude.

This plot is frequently normalized (by the sensitivity) to show deviations from a flat amplitude response.Figure 1 is an illustration of the amplitude response for an ideal second-order system.

The amplitude response contains a great deal of information relative to the transducer, such as resonantfrequencies, bandwidth, and the damping of the resonances.

4.3.2 Phase response

The phase of the transfer function versus frequency. It can be derived from Equations 4.2 and 4.3 bysubstituting j w for s and computing the phase, as w is varied. Figure 2 illustrates the phase response ofthe ideal second-order system.

In the time domain, phase influences the instantaneous shape of the response to an input signal andcontributes to a time lag in transducer response.

4.3.3 Resonant frequency w r

The measurand frequency at which a transducer responds with the maximum output amplitude. Mosttransducers have more than one resonance, and the lowest in frequency, or first resonance, is usuallyconsidered more important. If the first resonance is the dominant one, the second-order systemapproximation may be valid. The normalized amplitude response at resonance is governed by the amountof damping in the system (see Figure 1).

4.4 Properties in the time domain

The properties in the time domain are descriptions of the transducers response to a specified input,usually a step function.

4.4.1 Ringing frequency ( w d ) (sometimes referred to as damped natural frequency)

The frequency of free oscillations in the transducer output resulting from a step change in measurand.The ringing frequency is indicated by the number of oscillations per unit time. For the linear second-ordertransducer, the ringing frequency is related to the resonant frequency by

(Eq. 4.5) r22

d 211

w

z-

z-

=w

- 21 -

4.4.2 Damping

The energy dissipating characteristic that, together with natural frequency, determines the upper limit offrequency response and the response-time characteristics of a transducer. In response to a step-changeof measurand, an underdamped (periodic) system oscillates about its final steady value before coming torest at that value; an overdamped (aperiodic) system comes to rest without overshoot; and a criticallydamped system is at the point of change between the underdamped and the overdamped conditions.

4.4.3 Damping ratio ( z )

The ratio of the actual damping to the damping required for critical damping. In Equation 4.2, thecoefficient z is the damping ratio. Typical dynamic pressure transducers have damping ratios much lessthan unity; consequently values of w d and w r very nearly coincide. The damping ratio is definedspecifically for a linear second-order system. Transducers with more resonances are approximated byassociating a damping ratio with each resonant frequency.

The damping ratio is a useful parameter in both the time and frequency domain. In the time domain, z isrelated to the amount of overshoot (see Figure 2) and influences the number of ringing cycles presentafter a shock excitation. In the amplitude response, z is related to the height of the peak at the resonantfrequency.

4.4.4 Rise time

The length of time required for the output of a transducer to rise from 10% of its initial value to 90% of itsfinal value when excited by a step change in measurand. Rise time is related to transducer frequencyresponse.

4.4.5 Overshoot

The amount of output measured beyond the final steady output value in response to a step change in themeasurand. The maximum theoretical overshoot of an ideal second-order transducer is 100 percent; thisoccurs when z is zero. Overshoot is determined as

(Eq. 4.6)

z-

zp

-

=

21e100overshoot

for the condition z 0.1.

4.4.6 Settling time

The time required after the application of a step change in measurand for the transducer output to settlewithin a small specified percentage (5 percent) of its final value. For the ideal second-order transducerwith small z

(Eq. 4.7)d

2

s13

twz

z-

=

The settling time increases with smaller z and w d. The number of oscillations at w d required to settle within5 percent of final value for the ideal second-order transducer is

ISA-37.16.01-2002 - 22 -

(Eq. 4.8)zp

z-

=

213N

2

4.4.7 Discharge time constant (DTC)

Time required for a transducer or measuring system to discharge its signal to 37% of the original valuefrom a step-change measurand. It relates to low-frequency measuring capability for both transient andsinusoidal events.

5 Dynamic pressure generators

The dynamic calibration of pressure transducers requires that the measurand produced by a dynamicpressure generator varies in time in both a known and an appropriate manner. With some generators, thepressure-time relationship can be predicted quite accurately. With others, the pressure-time relationshipcan be established accurately only with the aid of comparison to referenced pressure transducers. Whilereproducibility is a highly desirable characteristic of the dynamic pressure generator, it is not an essentialcharacteristic. When such a characteristic is lacking in a generator, full dependence on the referencetransducer is required.

Dynamic-pressure generators fall into two basic classes: aperiodic and periodic. The aperiodic generatorsare characterized by the pulse shapes they produce, such as the step or the peaking pulse. Quick-opening valve devices and pulse generators produce pressure rise times generally in the milli-secondrange or less. The rise times and the pressure amplitudes generated by these devices vary markedly fromone type of aperiodic pressure generator to another. The shock tube, for example, is capable ofgenerating pressure steps having rise times in the nanosecond range. A number of the dynamiccalibrators described in this clause are now commercial products.

Pressure step, as used in this document, is defined as a change in measurand in which the rise time isless than one-fifth the rise time of the transducer measuring it.

Sinusoidal pressure generators, which require the use of a transfer standard, are the most useful of thevarious periodic pressure generators available, however, and these devices are limited as to useablerange of frequency dynamic pressure ratio and dynamic amplitude. Nonsinusoidal pressure generatorsof significant usefulness include the square wave or rectangular wave generators, which may beconsidered as a special case of the aperiodic or step-function generators. Figures 4 and 5 present asummary of the capabilities of the dynamic pressure generators.

5.1 Shock tube

A shock tube, in its simplest form, consists of two sections of tubing separated by a thin diaphragm. Whenthese two sections are pressurized to different pressure levels, and the diaphragm is suddenly ruptured,the higher-pressure gas will immediately begin to flow and compress the gas at a lower pressure(References 56, 57, 58).

It should be noted that most cold-gas, shock-tube-development work occurred in or before the 1960s.However, in 1997, a shock tube was designed and built at a university for a transducer manufacturer. Thedevelopment report for this new shock tube, Reference 64, also updates the literature through theintervening time period.

At a distance of approximately 10 to 15 tube diameters downstream from the diaphragm, a well-formedshock wave is established. This shock wave continues to move through the remainder of the gas in thelow-pressure section at approximately a constant velocity. Behind the shock wave, the pressure suddenlyrises to a new value, resulting in a positive pressure step. The length of time the pressure remainsconstant behind the shock wave depends on the dimensions of the shock tube, the position in the low-

- 23 -

pressure section at which the pressure is being monitored, the degree of smoothness of the inner walls ofthe low-pressure section, the type and design of the diaphragm, and the type, temperature, and initialpressure of the gas in each section. Air or helium and air in combination are commonly used gases.

When a shock tube is utilized for pressure transducer calibration, several parameters must be measuredbefore the amplitude of the pressure step can be ascertained. These parameters include the shock-wavevelocity, Vs, and the initial absolute pressure, P1, and temperature, T1, of the gas in the low-pressuresection.

5.1.1 Sidewall transducer mounting

If a pressure transducer is mounted flush in the sidewall of the low-pressure section, it will sense achange in pressure, D P, when the shock wave passes over it. This is commonly referred to as "incident"pressure. The equilibrium pressure and particle velocity behind the shock wave are determined from theRankine-Hugoniot relations (References 2, 6, 24, 25, 26, 27, 28, and 58). When air is used as the workinggas (low-pressure section), the amplitude of the pressure step can be computed from the followingequations:

(Eq. 5.1) ( )1MP67P 2s1 -=D

and

(Eq. 5.2)

+

=

1

ss T273

2985.344

VM

where Vs is expressed in meters per second, T1 is gas temperature in degrees, C, P1 is absolute pressure,and Ms is the shock-wave Mach number. When gases other than air are used, Equations 5.1 and 5.2 donot apply. (See References 2, 24, and 25 for further information.)

Since the working-gas temperature must be known, the convenient method of determining thetemperature is from a measurement of the static-wall temperature of the shock tube. Except at very lowpressures, the temperature of the working gas closely approaches that of the wall in less than oneminute.

The shock-wave velocity, Vs, is determined by measuring the shock-wave transit time between pointsspaced a known distance apart along the path of the velocity vector. Because the velocity of the shockwave tends to decrease with distance, the last pair of points should be in close proximity (less than 1 tubediameter) to the transducer undergoing calibration. Several different types of sensors are used to detectthe passage of the shock wave past these points; the most common being pressure transducers, thin-filmheat-transfer gages, and light screens. The shock-wave transit time, D ts, between pairs of sensors, ismeasured with digital timing. The shock-wave velocity, Vs, is calculated using the equation Vs = spacingbetween sensors/ D t s. Because of the squared relationship between Vs and D P, an uncertainty of 0.5percent in the measurement of the shock-wave velocity produces an uncertainty of at least 1 percent inthe determination of pressure-step amplitude.

When the pressure transducer is mounted flush in the sidewall of the shock tube, the rise time of thepressure step resulting from the passage of the shock wave is dependent on both the shock-wavevelocity and the transverse length of the transducer diaphragm, d, in the direction of shock-wavepropagation. Consequently, pressure transducers with the fastest rise time and smallest diameterdiaphragm should be used. Commercially available micro-sized pressure transducers with a 1mmeffective area reduce transient time across the diaphragm to 2 to 3 microseconds.

ISA-37.16.01-2002 - 24 -

The time required for the pressure on the transducer to change from P1 to P2 (P2 = P1 + D P) is given by theexpression

(Eq. 5.3)sV

dt =

Figure 4 Aperiodic generators

- 25 -

Figure 5 Periodic generators

ISA-37.16.01-2002 - 26 -

The maximum theoretical rise time, tr , for pressure transducers with circular diaphragms mounted flushin the sidewall of a shock tube can be shown to be

(Eq. 5.4)s

r Vd687.0t =

The side-mounted mode of operation is recommended

1) when this is the manner in which the transducer will be used in application;

2) when maximum accuracy in the determination of the pressure-step amplitude is desired;

3) when it is desirable to minimize transducer ringing; and

4) when the incident wave is considerably cleaner than the reflected wave.

5.1.2 End-wall transducer mounting

If the end of the low-pressure section of the shock tube is sealed off with an end plate, the moving shockwave, in striking the plate, will reflect from it. This is commonly referred to as a "reflected" shock wave. Apressure transducer mounted flush in the end plate will detect only the reflected shock wave. Thereflected shock wave is characterized by a much shorter rise time (usually nanoseconds) and a higherpressure as compared with the incident shock wave (sidewall measurement). The rise time of thepressure step associated with the reflected shock wave is sufficiently short to excite all the ringingfrequencies associated with virtually all flush-mounted pressure transducers. A tourmaline pressure bartransducer specially designed for reflected shock-wave measurements has an ~ 0.2 m sec rise time andnonresonant response (References 57, 58). When air is used as the work gas, the amplitude of thepressure step behind the reflected shock wave is

(Eq. 5.5) ( )

+

+-=D 2

s

2s2

s1 M5M421MP

37P

where Ms and P1 are defined as in Equations 5.1 and 5.2.

Because of the complex relationship between D P and Ms in Equation 5.5, any uncertainty in themeasurement of the shock-wave propagation Ms may produce an uncertainty in the determination of thepressure amplitude D P several times larger. (Reference 2 provides a convenient source of workingequations when gases other than air are used.)

The pressure behind the incident and reflected shock waves remains constant for a period of time, whichis dependent on the design of the shock tube and on the type, temperature, and pressure of the gasesinitially in the two sections. For a given shock-tube geometry, the longest duration of constant pressurebehind the reflected shock wave is obtained when the shock tube is operated under tailored conditions,as described in References 2, 10, and 26. Depending on the operating conditions and shock-tubegeometry, the period of constant pressure behind the reflected shock wave may vary from a few hundredmicroseconds to several milliseconds.

The end-plate mounted mode of operation is recommended

a) for the determination of transducer ringing frequencies;

b) when this is the manner in which the transducer is to be used in application;

- 27 -

c) when the maximum pressure step amplitude is required in calibration;

d) when the maximum duration of constant pressure behind the shock wave is desired; and

e) when determination of ringing frequency of gas passage is associated with transducer recess mount.

5.1.3 Other considerations

Acceleration of the walls and end plate of a shock tube occurs during operation of the device. In order todetermine the effect of this ground shock acceleration on the transducer output, the sensing end of thetransducer must be blanked off from the pressure wave without significantly altering the accelerationcomponents. Acceleration effects can be minimized by utilizing heavy-walled tubing for the low-pressuresection of the shock tube, by using a heavy end plate, and by shock-mounting the tube. Modern miniatureacceleration-compensated transducers are less susceptible to mechanical vibrations traveling along theshock-tube walls.

When a shock wave passes through the working gas, the temperature of the gas is suddenly raised. Thenew temperature, T2, varies with the square of the shock wave Mach number, Ms, as well as with the typeand initial temperature of the gas. If the transducer is sensitive to transient temperatures, then thetemperature step produced by the shock wave may cause errors in the transducer calibration.

To determine the extent of such errors and to reduce their effects, a temperature shield may beemployed. Thin coatings of opaque insulating materials sprayed on or bonded to the diaphragm makegood shields but may alter the transducer characteristics. A transducer should always be checked fortransient temperature sensitivity (7.3.3 provides further information).

5.1.4 Recommended shock-tube operating conditions

1) When the reflected shock-wave mode of operation is used, the shock tube should be operated undertailored conditions. (See description in References 2, 26, and 28.)

2) The test transducer should be flush-mounted in the sidewall or solid-end plate of the shock tubeunless special considerations indicate otherwise.

3) The shock tube must be kept free of diaphragm fragments, gas and oil contaminants, moisture, andthermal effects remaining from previous operations.

4) The significance of both acceleration (ground shock) and transient temperature on the response ofthe transducer must be investigated prior to making a pressure calibration.

5.2 Shockless pressure-step generators

A number of devices have been developed that generate a rapidly rising pressure step between twopressure levels (see References 2, 15, 16, 17, 27, and 58). Most of these units employ a quick-openingvalve. However, at least one utilizes a burst diaphragm. The geometry of the generator and the openingtime of the valve or burst diaphragm are such as to preclude the formation of shock waves when thedevice is operated. Shockless step generators have been designed and successfully used to produceboth increasing and decreasing pressure steps. Although most of these generators employ gaseousmedia, a few liquid-medium devices have been developed and used. The shockless pressure-stepgenerator, now commercially available, has the following advantages over other dynamic-pressurecalibrators:

a) The magnitude of the pressure step generated by the device is determined by measurements of staticpressure on the test transducer before and after the quick-opening valve is opened, therebypermitting high accuracy in its determination.

ISA-37.16.01-2002 - 28 -

b) The duration of constant pressure behind the pressure step can be made as long as desired.

c) Both the initial pressure on the transducer and the magnitude of the pressure step are controllableover very wide pressure ranges.

d) In general, it is superior to the shock tube from the standpoint of operational speed and simplicity oftechnique.

The dynamic characteristics of a shockless pressure-step generator are determined by measurement witha calibrated reference transducer possessing a rise time of no more than one-fifth that of the measurand.The following dynamic characteristics of the generator should be known: rise time, overshoot,undershoot, and the inherent ringing frequencies with their associated damping ratios. Also of interest isthe stability of both static-pressure levels P1 (initial pressure) and P2 (final pressure).

When the pressure rise time is one-fifth that of the transducer undergoing test or calibration, the error inthe measurand value of transducer rise time is less than 1 percent (see 6.8). If this criterion is not met, thecomplete rise time must be analyzed carefully for meaningful results.

Acceleration is present during the operation of the shockless pressure-step generator, and this should beminimized by design. In general, the shorter the rise time of the device, the greater is the level ofacceleration (ground shock). In those units that utilize poppet valves, it may be necessary to open thepoppet valve more slowly when calibrating at very low pressures in order to keep the acceleration level toa minimum.

Associated with the pressure step produced by these generators is a dynamic temperature change inwhich amplitude is related directly to the pressure change, P2 P1, and inversely to the rise time of themeasurand. As with shock tubes, the effect of the dynamic temperature pulse on the response of both thetest and reference transducers must be determined. When a gas medium is used in the shockless stepgenerator, the rise time of the measurand is inversely related to the speed of sound in the gas. For thisreason helium is used when very short rise times are desired.

The following calibration conditions are recommended:

1) The test transducer should be mounted flush in the generator with a minimum of volume between thevalve or diaphragm and the transducer diaphragm.

2) In the determination of transducer rise time and overshoot, the rise time of the generator should beless than one-fifth that of the transducer.

3) The amplitude of the pressure steps generated should cover the range of the transducer or range ofapplication.

4) The medium (gas or liquid) in the step generator should be similar to that used in the final application.

5) When possible, the generated pressure steps should be of the same direction as encountered in theapplication, i.e., either increasing or decreasing pressure.

6) The significance of the transducers response to acceleration and to the dynamic temperature pulsein the generator should be determined.

5.3 Pulse generators

Several devices have been developed to provide single-peaking pulses of reasonably controlledamplitudes. These pulse generators produce a dynamic measurand that is not a step function, but thatmay resemble a single half cycle of a sine wave. One technique employed to generate such a pulse is to

- 29 -

drop a mass onto a piston in contact with the surface of an incompressible fluid contained within a fixedvolume (References 14, 66, and 73). The commercial version of this is referred to as a hydraulic-impulsecalibrator. The device consists of a piston/cylinder manifold and a drop tube containing a mass that canbe dropped onto the piston from various heights. The amplitude of the pulse is dependent on the fluidincompressibility, the mass, its initial height above the piston, and the piston area. The pulse generator isnot an absolute calibration device and requires a comparison pressure transducer of knowncharacteristics to monitor the pulse and provide a peak value measurement for the test transducer.Alternately, commercial versions that operate to 100,000 psi depend on acceleration references on aknown mass (References 55, 64).

The greatest advantage of the pulse generator is the comparative ease with which very high-pressurepulses can be generated. Care must be taken in the selection and location of the reference transducerused since results of the calibration are dependent on this comparison standard. Tourmaline transducers,which are volumetrically sensitive, are commonly used as transfer standards in hydraulic-impulsecalibrators (Reference 57). Hydrostatic pressure is applied directly to the crystal. The recommendedconditions of operation of 5.2 relative to the comparison transducer apply equally well to thesegenerators. In order to achieve accuracy in calibration using the pulse generator, it is essential that nopockets of gas exist at the diaphragm of either the comparison or test transducers.

5.4 Periodic pressure function generators (sinusoidal pressure generators)

The dynamic calibration of a pressure transducer could ideally be accomplished by sensing known inputsfrom a periodic pressure generator at known frequencies and amplitudes if such a device existed. Theobserved response, including the magnitude, waveform, and phase lag could then be compared with theknown input at various conditions. In order to calibrate with only one frequency at a time for accuracy andsimplicity, a sinusoidal pressure generator (SPG) is required. In practice, there are limitations to thisapproach. First, the applied average pressure levels and dynamic amplitudes generally are not known byabsolute means, and must be measured by another transducer. The SPG generates a pulsating pressurein a small chamber that can be monitored simultaneously both by a reference standard transducer and bythe transducer being calibrated (References 21, 57). The two transducers must be sufficiently close sothat they sense the same pressure, including amplitude, shape, and phase lag. Analysis of the output ofthe transducer being calibrated is thus entirely dependent on the performance of the reference transducerand what is known about this performance. The reference transducer, if statically calibrated, should alsobe calibrated by dynamic methods to establish that its sensitivity derived from static and dynamiccalibration is the same. Credibility of the dynamic sensitivity of the reference transducer is a basiclimitation of SPG utilization to a comparison process. As long as the reference transducer is provided withcredible dynamic calibration, it may not be a serious limitation because high-quality reference transducerscan be selected that have response characteristics exceeding the pressure, pressure amplitude, andfrequency that can be obtained with available SPGs. Otherwise, it may be difficult to present a compellingargument concerning the validity of any calibration that uses a statically calibrated transducer as areference standard for dynamic calibration.

The governing limitations are associated with the ability of the SPG to provide the desired signal. An SPGdevice, when used for calibrating a pressure transducer for a specific use, should satisfy the following:

a) The pressure generated is sinusoidal such that frequencies other than the fundamental are negligible.

b) The frequency range generated covers the frequencies of pressure expected in the intendedapplication.

c) The operating pressure range covers the transducer rating and/or intended application.

d) The dynamic pressure amplitude generated is large enough to identify possible nonlinearities in thetransducer amplitude response.

ISA-37.16.01-2002 - 30 -

e) The SPG is operated with the same medium (gas or liquid) with which the transducer is to be used.

In many cases, these criteria cannot be met, and a less-than-desired match is obtained between thedynamic pressure measurand applied during calibration and that encountered in use of the instrument.

Many special devices have been proposed and developed as SPGs, and these are described inconsiderable detail (References 2, 18, 19, 20, 21, 22, 23, 67, and 71). The SPGs can be categorized asacoustic resonators, variable-volume generators, or variable-mass generators. Little has been done tofurther develop SPGS, since the 1960s when sinusoidal calibrator research was government funded.

5.4.1 Acoustic resonators

Any of several driving devices can be used to establish acoustic resonance within a chamber. Thefrequency of resonance is fixed by the geometry of the chamber and the properties of the working fluid.To obtain a specific frequency, the chamber length must be a multiple of the acoustic half wavelength.When another frequency is required, the geometry must be changed, or harmonics of the fundamentalfrequency must be used (References 20 and 23).

At resonance, the pressure waves may be distorted to a significant degree from the pure sinusoidalexcitation because of gas dynamic phenomena. As the amplitude or frequency is increased, thenonlinearities associated with real gas, wall, and friction effects also become significant. At low dynamic-pressure amplitudes, these generators can provide pressure pulses that are sufficiently sinusoidal formany uses.

5.4.2 Variable-volume generators

In the variable-volume generator, a relatively fixed mass of working fluid is alternately compressed andexpanded within a small chamber. The chamber is deliberately made small such that its naturalfrequencies are always higher than the frequencies imposed, and thus resonant effects are eliminated. Apiston or diaphragm driver is used to vary the chamber volume and thereby the pressure in a repetitivemanner (Reference 11, 52). This methodology has been mostly used in the development of soundpressure calibrators.

Usually the gas compression is isentropic, and the pressure, p, follows the piston position, l, as follows:

(Eq. 5.6)g

=

l

lo

opp

where po is the equilibrium pressure and lo is the driver (piston or diaphragm) position at the equilibriumpressure, and g = ratio of specific heats of constant pressure and volume (Cp /Cv). Thus, as the pistontravels in a sinusoidal manner, the pressure amplitude is represented by the expression

(Eq. 5.7) ( ) g-wa+= tsin1pp oand the dynamic pressure amplitude by the expression

(Eq. 5.8) ( ) ...tsin1p2

tsinp])tsin1(1[pppp 2o2

ooo w+gga

-wag=wa+-=-=D g-

where a = modulation factor, which is always < 1.

Considering subsequent terms in the expansion with realistic coefficients, this dynamic pressure is clearlynonsinusoidal. The effects of fluid motion and viscosity also introduce nonlinearities in the wave form.

- 31 -

This class of generators is generally limited by the amplitude-frequency characteristics of the driving unit.In theory, nonsinusoidal input from the driving device could compensate for nonlinearities and thus asinusoidal pressure-pulse shape could be approximated. However, the dynamic characteristics ofpractical driver systems (e.g., crystal diaphragm or electromagnetically driven piston) naturally degradeinto sinusoidal displacements as the frequency is increased, thus limiting the effectiveness of thisapproach.

5.4.3 Variable-mass generators

The variable-mass generators utilize a fixed chamber volume, and the rate of fluid flow into or out of thechamber is cyclically varied to develop the dynamic pressure pulsations. These flow-modulated devicesprovide a fast response so that relatively larger pressure amplitudes are available at high-frequencyconditions compared to the variable-volume generators. The critical frequency limitation associated withacoustic dimensions of the chamber still apply, and operation is limited to frequencies appreciably belowthe natural frequency of the chamber, which is dependent on the properties of the fluid used and thechamber dimensions. It should be noted that these devices have not been commercially produced.

A siren-type device is often used for this class of SPG with the gas entering the chamber through acritical-flow orifice or nozzle from a constant pressure source and leaving the chamber through a largercritical-flow nozzle (References 22 and 23). The throat of one of these nozzles is interrupted by a rotatingdisc or cylinder with throat-sized holes. The pressure in the chamber is dependent on both the rate ofmass addition and the rate of mass discharge.

(Eq. 5.9) ss

ei pppAA

dtdp

-h=

where

(Eq.5.10) ( )121

12

v

a -g+g

+g=h = constant

a = gas speed of sound

v = chamber volume

g = ratio of specific heats of constant pressure and volume

A i = inlet orifice area

A e = outlet orifice area

ps = stagnation pressure of supply gas.

The dynamic pressure component, D P, inside the chamber experienced by both the test and referencetransducers is given by the following expression when the exit area is ( )( )tcos12/AA ee w+= :(Eq. 5.11)

w+w

w

h+w

w

h-=D t2cos41tcos

2A

tsin2pAp ee

where eA = maximum exit area and p = average chamber pressure.

ISA-37.16.01-2002 - 32 -

In order that the dynamic pressure component varies sinusoidally, the following conditions must be met:

e

i

AA2p =

and

12A

pP e

- 33 -

j) The working fluid used in the generator should be the same state (liquid or gas) as in the intendeduse of the transducer being calibrated.

k) The reference transducer should be calibrated by other than static methods (i.e., shock-tube and/orstep-pressure generator) to establish dynamic sensitivity and frequency response characteristics.

As noted in 5.4, under periodic pressure-function generators, in many cases the above criteria cannot bemet and a less-than-desired match between the periodic calibration and measurand requirements can beachieved.

6 Measurement of transducer properties

6.1 Sensitivity

In a transducer sensitivity measurement, either periodic or aperiodic pressure generators may be used toproduce the measurand. It is preferable to use a generator for which the dynamic pressure amplitude canbe accurately established without use of a dynamic reference transducer. To a limited extent, the shocktube satisfies this requirement. Today, calibration shock tubes with more precise measurement capabilityfor shock velocity and pressure amplitude are achieving uncertainties approaching 2%. The shocklesspressure-step generators (e.g., quick-opening valve devices) expose the transducer being calibrated to aprecisely known static pressure in about 50 m sec (Reference 59). A fixed-displacement piston-phone,commonly used for calibrating microphones, allows precise sensitivity measurements at low frequencieswithout a reference transducer, but only at very low-pressure levels. If the transducer responds to staticpressures, the typical static-pressure generators (such as hydraulic dead-weight testers) can be readilyused to establish the static sensitivity.

When used with a step-pressure source, a transducer with less than critical damping will produce anoscillatory output. In a shock tube, the reflected wave may disturb the transducer output before thetransducer oscillations decay. In this case, the average value of the oscillations must be estimated inorder to determine the sensitivity. In general, quick-opening valves allow application of an undisturbedpressure long after the oscillations decay.

If the transducer does not respond to static pressures, waiting for the oscillations to decay can contributeto an error in measurement of sensitivity. For example, a transducer with a single RC roll-off at 1 Hz(described by Equation 4.3) has dropped 5 percent of its value approximately 8 milliseconds after theapplication of a pressure step. In this case it requires that the oscillations be averaged in the first fewmilliseconds for such a transducer. Sinusoidal pressure generators can be used for straightforwarddetermination of sensitivity; however, precise pressures, as measured by a transfer standard transducer,can be generated only at low-pressure levels and relatively low frequencies.

Most of the properties defined for transducers indicate the magnitude of sensitivity variation with theseconditions, e.g., variations in response and linearity. It is therefore recommended that measurements todetermine the sensitivity be made under user-operating conditions.

6.2 Amplitude response

Amplitude response is one of the most important (but more difficult to obtain) properties of a transducer.Ideally, this measurement is performed with a sinusoidal pressure generator, which is swept over thefrequency range, yielding a constant dynamic-pressure amplitude at each frequency. Unfortunately, anSPG, which covers the amplitude and frequency range for most dynamic transducers, does not exist. Ingeneral, a flat frequency response (constant amplitude) cannot be guaranteed from the sinusodialpressure generator; therefore, the pressure generated must be monitored by a reference transducer,which should have sensitivity documented through dynamic calibration techniques. The ratio of theresponse of the transducer under test to that of the reference transducer is recorded. The naturalfrequency of the reference transducer must be at least five times the measurand frequencies.

ISA-37.16.01-2002 - 34 -

It is difficult to generate sinusoidal pressures at frequencies as high as the first resonance of mostdynamic transducers. Today, dynamic-pressure transducers have frequency response to 500 KHz andsome to >1 MHz. This has led to the use of aperiodic generators (such as the shock tube) to measure thetransducers amplitude response.

6.2.1 Amplitude response measurements by sinusoidal pressure

Limitations of sine pressure generator measurements are discussed in this subclause. The referencetransducer must be located very close to the transducer under test so that the same pressure field is seenby both transducers. A general rule is that the distance separating the transducers be less than a tenthwavelength of the pressure wave. The pressure wavelength is

(Eq. 6.1)fa

=l

For air, the speed of sound, a is 1087.4 ft/sec at 0C and at one standard atmosphere, which results in awavelength of approximately 1.3 inches at 10,000 Hz. The value a increases as the square root of theabsolute temperature and is essentially independent of pressure.

The geometry of the setup is critical for high-frequency measurements. Typical sinusoidal pressuregenerators operate into a sealed cavity. The dimensions of this cavity must be such that it does not haveresonances in the frequency range of interest. For reasonable data, the first half-wave cavity resonanceshould occur at least five times higher than the highest test frequency. For air at room temperature 0.1-inch cavity length would resonate at approximately 33,000 Hz, indicating a usable frequency range to6,600 Hz. It must also be established that the pressure wave is essentially a plane wave; that is, thatresonant modes do not exist across the cavity.

The usable frequency of a cavity is often increased by changing the media from gas to liquid. The soundpropagation velocity C in many liquids is approximately five times that in air. This increases the usablefrequency of the 0.1-inch cavity to 33,000 Hz, assuming a cavity with rigid walls and fluids with no gasbubbles.

If the ambient environment changes from a gas to a liquid, a question is raised about the validity ofcalibration with a liquid of a transducer to be used in a gas. Equation 1.2 shows the transducers transferfunction to be dependent on the transducer parameters of stiffness, mass, and losses. It is apparent thatthe equivalent moving mass would be greater for a liquid than a gas, and the losses could be different.Changes in these coefficients may alter the resonant frequency and the damping ratio considerably.

Harmonic distortion in the sinusoidal pressure generator can indicate false minor resonant frequencies inthe amplitude response. For example, if a resonance exists at f1 (either from transducer or cavity), and thegenerator being used has a strong nth harmonic distortion, a minor resonance may falsely appear whenthe generator is at f1 / n. For this to occur, the resonant amplification times the n

th harmonic must be equal

to or greater than the fundamental; for example, 20 dB resonance and a 10 percent second harmonic. Ifthis effect is suspected in a measurement, the minor resonances can be examined by rejecting theharmonic in the transducer output by the use of a tunable filter. In this way it can be demonstrated thatminor resonances are valid and not due to a testing error.

The influence of distortion of the pressure waveform on the accuracy of the amplitude response dependsupon the method of reading the output signals. For example, if the peak-to-peak value of theinstantaneous signal is taken from a digital recorder, distortion and noise can disturb the shape of thewave and make the readings difficult. However, if root-mean square or rectified-average readings areused, small perturbations on the instantaneous signal are not significant. For this reason rms or rectifieddetection is recommended for all steady-state sinusoidal measurements.

- 35 -

It is sometimes necessary to acquire response data at pressure levels that are low relative to thetransducers full-scale capability. In this case, electrical noise may be a problem, and it may be necessaryto use a filter. The characteristics of this filter may be determined in the following manner by inserting avoltage at the transducer at the same frequency. A low resistance is inserted in the ground return lead ofthe transducer, and an accurate AC voltage is applied across the resistor from an ungrounded oscillator.The insert voltage-vs.-frequency characteristic can be established with high accuracy, and this is used tocorrect the pressure data to yield the curve for the transducer alone. As an alternative, the entire system(transducer and filter) can be calibrated as a unit using the techniques previously outlined.

6.2.2 Amplitude response with aperiodic sources

The shock tube can be used to determine the amplitude (and phase) response of a transducer bymathematically transforming the pressure input and transducer output from the time domain into thefrequency domain. The theory for the mathematical operations is fully developed and can be quicklyperformed with digital signal-processing instrumentation (Reference 64).

It is typically assumed that the pressure input to the transducer approximates a step function. A referencetransducer is not used. The response to the step function depends upon whether the test transducer issubjected to reflected measurements at the end or along the side of the shock tube. The rise time of thereflected pressure at an end-mounted transducer is very fast, but a side-mounted transducer sees theincident shock wave pass across its diaphragm, resulting in a much slower rise time. In this case thesensitivity of portions of the diaphragm become significant, and the fine structure of the input is difficult todetermine. The frequencies determined by this technique may be limited by the pressure rise time of theshockwave and the duration of the pressure plateau.

In dealing with the transient response, care must be taken that other inputs (from temperature orvibration) do not contribute significantly to the transducer output. Otherwise transducer-mountingresonances may appear in the pressure data (see Clause 5). The geometry of inlet coupling portsassociated with recess-mounted transducers also affects their amplitude and phase response.

The time between samples limits the upper frequency that can be resolved. Typically, experimentersprefer 5 or more samples per sine wave for wave-shape definition. For example, a data sample every fivemicroseconds would allow wave-shape approximation to 40,000 Hz. However, frequency informationwould be preserved to 100,000 Hz based on Nyquist sampling. Given a transducer response that is cleanfor five milliseconds, a total of 1000 data points would be generated.

6.3 Phase response

Phase response is determined with a sinusoidal pressure generator by comparing the transducer outputvoltage waveform with that of the reference sensor, which is simultaneously excited by the periodicmeasurand. The apparatus is much the same as for amplitude response measurements, except thephase difference between the two waveforms is determined by means of a phase meter, Lissajouspatterns, or by accurately measuring the time shift between the two wave forms of a digital recorder; if notdual beam, operate in chop, not alternate, mode.

The reference-pressure transducer should have negligible phase shift in the frequency range in which thetest transducer is to be measured. Figure 2 shows the phase shift for second-order systems with variousdegrees of damping.

The positioning of the transducers in the measurement cavity is more critical for phase measurementsthan for amplitude measurements. A criterion of placement of l /10 was suggested for amplitudemeasurements, but this is equivalent to a 36-degree phase shift. Hence the criteria for phase-shiftmeasurements should be increased to l /50 to insure phase differences of no greater than 5 degrees. Thisis a very stringent requirement and essentially limits phase measurements to low frequencies.

ISA-37.16.01-2002 - 36 -

A technique for circumventing these criteria is to place the reference and test transducers at mirrorimage locations within the cavity. The use of two reference transducers is required to establish to whatextent the locations are mirror images.

In general, electronic filters are not recommended for use in phase measurements unless the cut-offfrequencies are a decade above the measurement frequencies, or unless the filters used for thereference and test transducers are matched in phase. The use of the insert voltage techniques, todetermine the validity of the method, is recommended if filters are used (Reference 10).

Phase measurements can also be mathematically derived from shock-tube measurements as a by-product of the amplitude-response data.

6.4 Resonant frequency

The transducer-resonant frequency is best determined from the transducers amplitude response,obtained with a reflected shock-tube measurement. Sinusoidal generators do not have high enoughfrequency to excite the resonant frequency of most dynamic-pressure sensors. It is readily determined byexamination as the frequency at which the transducer responds with maximum-output amplitude.Resonant frequency can be measured quickly and accurately using computer-data-processing software.The resonant frequency ( w r) can also be calculated from the ringing frequency ( w d) by use of therelationship

(Eq. 6.2) d22

r 121

w

z-

z-

=w

Because the resonant frequency is a function of damping ratio as well as natural frequency, the type ofpressure medium is important since the damping effect of a liquid medium can significantly affect theresonant frequency.

6.5 Ringing frequency

The response of an underdamped transducer to a step or impulse is a damped oscillatory transient orringing. The ringing frequency can be determined by applying a pressure-step input, usually from areflected shock wave to the transducer, recording the output and counting the cycles-per-unit time of thetransducers response. Typically, these data are quickly recorded and analyzed on a high-frequencydigital recorder. When a transducer exhibits more than one ringing frequency, the output is a combinationof these frequencies, and the measurement is determined with computer software. In this case thevarious ringing frequencies must be separated by means of filters before Equation 6.3 can be applied. Itcan be shown that the frequency response of such a system, below the lowest ringing frequency, is quitesimilar to that of a single-degree-of-freedom system.

6.6 Damping ratio

The damping ratio ( z ) can be obtained from the amplitude response using a sine-wave generator or canbe measured using an aperiodic generator.

The amplification factor (A r) of a resonance on the amplitude-ratio curve is related to the damping factor(assuming a linear second-order system)

(Eq. 6.3) 1707.0;1707.00;12

1A2

r z=z

z-z

=

and is plotted in Figure 1. Solving for z :

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(Eq. 6.4)2

A/111 2r2 --=z

If a step input is used to excite the transducer, the damping ratio can be determined from the followingrelationship (assuming a second-order system):

(Eq. 6.5)

2/12

2

110 V

Vlog303.2

N21

-

D

D

p

+=z

where N represents the number of complete cycles at a specific ringing frequency over which themeasurement is made; D V1 and D V2 are the peak incremental voltages above the average value at thebeginning and end of N cycles.

The accuracy of this method depends on the ability to establish the average value of voltage V , sinceVwhereVVV -=D is the peak voltage for any cycle. Because of the uncertainty in V , the greatest

accuracy in the determination of z is achieved when N equals 1. Equation (3.6) then becomes

(Eq. 6.6)

2/12

2

110 V

Vlog

728.21

-

D

D

+=z

6.7 Rise time

Rise time is measured by applying a step input to the transducer and measuring the time required foroutput to go from 10 percent to 90 percent of the final average value. For values of damping ratio of 0.5 orless, the rise time of the step-pressure input must be less than one-fifth that of the transducer for thetransducers rise time to be within 1 percent of its asymptotic value. So long as the rise time of the steppressure input is less than one-fourth that of the transducer having damping ratio of 0.1 or less, the risetime of the transducer will be within 1 percent of that obtained with a step-function (zero rise time) input.Care must be exercised that the rise time of the recording system is sufficiently short to introducenegligible error in the measurement.

ISA-37.16.01-2002 - 38 -

6.8 Overshoot

Overshoot is measured by observing the transducers response to a step input of pressure. Theovershoot is determined as the peak output (Vp) minus the average output ( V ) divided by the averageoutput

(Eq. 6.7)

-

=

VVV

Overshoot p 100 in percent

The maximum theoretical overshoot that a linear second-order system can have is 100 percent. Thisoccurs when the damping ratio is zero. The maximum overshoot is strongly dependent on both thedamping ratio and the quantity tw d where t is the input rise time, and w d is the ringing frequency of thetransducer. Most pressure sensors have damping ratios less than 0.1. Acceleration-compensatedpressure transducers have substantially less overshoot than non-compensated transducers. In addition,their overshoot is not linear with step-pressure amplitude. Generally, but not in all cases, their percentageovershoot will increase with larger step-pressure amplitude.

7 Transducer interfaces

The following four main factors need to be considered when installing dynamic-pressure transducerseither for calibration or for performing a measurement:

a) Strain effects

b) Cavity or passage resonances

c) Temperature effects

d) Acceleration effects

Other effects, such as earth-gravity field and those from the earth's magnetic field, which may besignificant when dealing with larger, more delicate mechanical instruments, will usually not affect thedynamic behavior of an electrical-pressure transducer but should be checked for in some cases. Whencalibrating vibration transducers on a shake table, spurious effects induced by the strong variablemagnetic field of the table are sometimes encountered. Again, such effects are seldom significant indynamic calibration of pressure transducers and will not receive consideration in this clause.

7.1 Mounting, strain effects

Both non-precision mounting and over-torquing induce strain into a transducer housing (Reference 73)and can be a source of measurement error. Strain introduced into the transducer body may manifest itselfeither as a change of the sensitivity, an increase in mechanically induced noise, or as a null shift. Strainsensitivity shift is normally noticed when calibration data varies in slope, depending on the mountingtorque applied. In order to assess this effect, the transducer should be calibrated first using therecommended mounting torque and then repeated with some specific over-torque and under-torquevalues, respectively. Calibrating a torque-sensitive transducer satisfactorily is a difficult task since otherfactors such as concentricity of the mounting hole, tightness of the thread, dirt particles, etc., may affectthe sensitivity and thus yield inconsistent results, which are hard to correlate.

The null shift (zero shift) caused by mounting strain represents the component of the transducer signal,which does not depend on the input pressure, but which is a shift in location of the calibration data curve.

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A torque-sensitive transducer acts like a strain gage. In a dynamic calibration or measurement, it willgenerate a spurious signal that reflects strain fluctuations in the mounting structure. No standardtechnique has been defined yet to accurately determine the magnitude of this effect in pressuretransducers. A simple method of detecting significant strain sensitivity is to connect the transducer to itsrecording equipment and check the change of the output while tightening it to its recommended mountingtorque in its mounting hole. This technique is useful only if the system has a good low-frequencyresponse. Manufacturers installation drawings should be followed closely and only the mounting partsshould be machined according to dimensions provided.

Transducers should be installed using the recommended mounting torque. Care must be taken in orderto avoid strains in the mounting structure that can affect the transducer performance during calibrationand use.

7.2 Cavities and passages

The manner in which a transducer is mechanically coupled to the pressure can significantly affect theresponse of that transducer. Meaningful measurements of pressure fluctuations at frequencies around10,000 Hz or higher can only be made with transducers having flush diaphragms. The use of anyconnecting line or cavity will limit the frequency characteristics of the measurement system itself.

There are instances where a connecting line or passage cannot be avoided. In such a case, its length willhave to be selected to be consistent with the highest frequency to be measured. The lowest longitudinalresonant frequency of a cylindrical passage is

(Eq. 7.1)L4af =

where a is the speed of sound in the gas at the given temperature and L is the length of the passage.

If a dynamic measurement or calibration has to be performed through a passage, the highest frequencyconsidered should be less than 1/10 of the resonant frequency of the passage. In air at room temperature(a = 1087.5 feet per second) and a passage length of 1/4 inch, for instance, frequencies up to 1400 Hzwill result in less than one percent dynamic error. This relationship applies only for straight passages. Asmall passage leading to a cavity in front of the transducer will result in much lower-resonant frequencies.The following relation shows the effect of a gas-filled passage and a cavity

(Eq. 7.2) [ ] 2/1V)d85.0L(09.7

cdf -+= providing p

>V4Ld2 the cavity resonance approaches the longitudinal resonant frequency described by

Equation (7.1).

The determination of all mechanical resonances basic to the transducer over the frequency range to bemeasured is important. The nature of these resonances may be somewhat obscure, and care must beexercised to insure their repetition from one installation to the next if a response analysis is to be valid.

ISA-37.16.01-2002 - 40 -

Resonances other than the major ringing frequency may sometimes be caused by non-flush diaphragms,discontinuities in the surface near the transducer, and vibration, etc. In short, modulating frequenciesabove or below the ringing frequency may not be inherent to the transducer at all. The presence of theseassociated resonances emphasizes the fact that if evaluations of transducer response are to bemeaningful, the mounting configuration employed for the calibration must be identical to that used in theactual application. Liquid-filled systems may introduce additional measurement uncertainties due tocavitation and inertial effects that are not readily predictable.

7.3 Temperature effects

Because of the temperature sensitivity of many dynamic-pressure transducers, temperature effects needto be considered in most applications and in calibration as well. When calibrating a temperature-sensitivetransducer with compressed air, for instance, the small temperature rise due to adiabatic compressionmay be sufficient to significantly distort the results. There are two basic effects due to temperature, thetemperature-sensitivity shift and the temperature-null shift. Furthermore, unevenly distributed (i.e.,transient) temperatures may cause quite different effects from those obtained with the transducer heateduniformly in a lab oven (Reference 31).

Accurate calibration of a dynamic transducer can only be obtained after its temperature sensitivity hasbeen assessed. If that should turn out to be excessive, temperature should be maintained constant at apredetermined value during calibration.

There are at least two methods for evaluating the transient thermal sensitivity of pressure transducers.The easiest method is to make a test measurement with and without ablative coating applied to thetransducer diaphragm (see 7.3.3). This may be practical if the cost factors involved with the testmeasurement are not significant and the data is repeatable, as might be with periodic co