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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Co ntents
ArticlesSpecial Report on Electrical StandardsNew
Internationally Adopted Reference B. N. Taylor 95
Standards of Voltage and Resistance
A Supercritical Fluid Chromatograph Thomas J. Bruno 105for
Physicochemical Studies
Relation Between Wire Resistance and Fluid Pressure H. M. Roder
and 113in the Transient Hot-Wire Method R. A. Perkins
Scattering Parameters Representing Imperfections Donald R. Holt
117in Precision Coaxial Air Lines
Conference ReportsFourth International Symposium on J. P. Young
135
Resonance Ionization Spectroscopyand Its Applications
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Departments
News Briefs
DEVELOPMENTSConductivity Measurements on Insulating
FoamsDevelopment of Standard Weld ProceduresCharacterization of
Asphaltic CementsStructure Modulated Chromium Enhances Wear
PerformanceDiscovery of Pervasive Antiphase Boundaries in Liquid
Encapsulated
Czochralski-Grown Semi-Insulating Undoped Gallium
ArsenideIntercomparison of Radiometric Standards in the Near
InfraredNew Calculations of Inelastic Mean Free Paths for
Low-Energy Electrons in SolidsOptical Probes Developed for
Electromagnetic Field MeasurementsToo Hot to Handle, but Not to
MeasureNIST/IBM Neutron Reflection Studies of Polymer Surfaces and
InterfacesDiamond Films ExaminedSuccessful Validation by the Key
Management Validation SystemNew Probe Characterization Technique
Promotes More Efficient Use of Geostationary
Orbit Frequency Space by Communications SatellitesVolt, Ohm
Standards to Change Internationally in 1990Test Instrument to
Detect Computer Flaws PatentedBetter Sheet Metal Products with Less
WasteMaking Invention Pay1989 National Quality Award Applications
IssuedShedding New Light on Ways to Cut Energy DollarsTest Can Help
Ensure Industrial Chemical PurityNew Calibration Services
OfferedPapers Available on Optical Fiber MeasurementsCharacterizing
TEM Cells
137
STANDARD REFERENCE DATA 143CRYSTDAT: An Online Research and
Analytical ToolMajor Expansions Announced for Mass Spectral
Database
CALENDAR 144
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Special Report on Electrical Standards
New Internationally Adopted ReferenceStandards of Voltage and
Resistance
Volume 94 Number 2 March-April 1989
B. N. Taylor This report provides the background representation
of the volt be increasedfor and summarizes the main results of by
about 9.26 parts per million (ppm)
National Institute of Standards the 18th meeting of the
Consultative and the value of the U.S. representationand
Technology, Committee on Electricity (CCE) of the of the ohm be
increased by about 1.69Gaithersburg, MD 20899 International
Committee of Weights and ppm. The resulting increases in the
U.S.Measures (CIPM) held in September representations of the ampere
and watt
1988. Also included are the most impor- will be about 7.57 ppm
and 16.84 ppm,tant implications of these results. The respectively.
The CCE also recom-principal recommendations originating mended a
particular method, affirmedfrom the meeting, which were subse- by
the CIPM, of reporting calibrationquently adopted by the CIPM,
establish results obtained with the new referencenew international
reference standards of standards that is to be used by all
na-voltage and resistance based on the tional standards
laboratories.Josephson effect and the quantum Halleffect,
respectively. The new standards, Key words: CCE; CIPM;
Consultativewhich are to come into effect starting Committee on
Electricity; InternationalJanuary 1, 1990, will result in improved
Committee of Weights and Measures;uniformity of electrical
measurements International System of Units; Josephsonworldwide and
their consistency with effect; Josephson frequency-to-voltagethe
International System of Units or SI. quotient; ohm; quantum Hall
effect;To implement the CIPM recommenda- quantized Hall resistance;
SI; volt.tions in the U.S. requires that, onJanuary 1, 1990, the
value of the U.S. Accepted: December 7, 1988
1. Background
The 18th meeting of the Consultative Committeeon Electricity
(CCE) of the International Commit-tee of Weights and Measures
(CIPM) was heldSeptember 27 and 28, 1988, at the International
Bu-reau of Weights and Measures (BIPM), which islocated in Sevres
(a suburb of Paris), France. NISTDirector E. Ambler, a member of
the CIPM andPresident of the CCE, chaired the meeting and theauthor
attended as NIST representative. Some 30individuals from 15
countries participated.
As discussed in this journal in the author's 1987report on the
17th meeting of the CCE held at theBIPM in September 1986 [1], the
CCE is one of
eight CIPM Consultative Committees which to-gether cover most of
the areas of basic metrology.These Committees give advice to the
CIPM onmatters referred to them. They may, for example,form
"Working Groups" to study special subjectsand make specific
proposals to the CIPM concern-ing changes in laboratory reference
standards andin the definitions of units. As organizational
entitiesof the Treaty of the Meter, one of the responsibili-ties of
the Consultative Committees is to ensure thepropagation and
improvement of the InternationalSystem of Units or SI, the unit
system usedthroughout the world. The SI serves as a basis for
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the promotion of long-term, worldwide uniformityof measurements
which is of considerable impor-tance to science, commerce, and
industry.
However, scientific, commercial, and industrialrequirements for
the long-term repeatability andworldwide consistency of voltage and
resistancemeasurements often exceed the accuracy withwhich the SI
units for such measurements, the volt'and the ohm, can be readily
realized. To meet thesesevere demands, it is necessary to establish
repre-sentations' of the volt and ohm that have a long-term
reproducibility and constancy superior to thepresent direct
realizations of the SI units them-selves.
Indeed, as discussed by the author in reference[1], in 1972 the
CCE suggested that the nationalstandards laboratories adopt 483 594
GHz/V ex-actly as a conventional value of the Josephson
fre-quency-to-voltage quotient for use in maintainingan accurate
and reproducible representation of thevolt by means of the
Josephson effect. While mostnational laboratories did adopt this
value, three de-cided to use different values. Moreover, it has
be-come apparent that the CCE's 1972 value of thisquotient is about
8 parts per million (ppm) smallerthan the SI value, implying that
representations ofthe volt based on the 1972 value are actually
about8 ppm smaller than the volt.
It has also become apparent that because mostnational standards
laboratories base their represen-tation of the ohm on the mean
resistance of a par-ticular group of wire-wound resistors, the
variousnational representations of the ohm differ signifi-cantly
from each other and the ohm, and some aredrifting excessively.
Although the Thompson-Lampard calculable capacitor can be used to
real-ize the ohm with an uncertainty 2 of less than 0.1ppm, it is a
difficult experiment to perform rou-tinely. Hence, the 1980
discovery of the quantum
'The volt is the SI unit of electromotive force (emf) and
electricpotential difference. Occasionally it may be referred to in
theliterature as the absolute volt. As-maintained volt,
representa-tion of the volt, laboratory representation of the volt,
"nationalunit of voltage", "laboratory unit of voltage", "practical
realiza-tion of the volt", and other similar terms are commonly
used toindicate a "practical unit" for expressing measurement
results.However, to avoid possible misunderstanding, it is best not
touse the word unit in this context. The only unit of emf in the
SIis, of course, the volt. In keeping with references [2] and
[3],from which this report has drawn heavily, we use the
expres-sion representation of the volt and variations thereof. The
expres-sion reference standard of voltage is also used occasionally
in asimilar or related sense. The situation for the ohm and
resistanceis strictly analogous.2 Throughout, all uncertainties are
meant to correspond to onestandard deviation estimates in keeping
with CIPM Recommen-dation 1 (CI-1986) [4,5].
Hall effect (QHE) by K. von Klitzing [6] was en-thusiastically
welcomed by electrical metrologistsbecause it promised to provide a
method for basinga representation of the ohm on invariant
fundamen-tal constants in direct analogy with the Josephsoneffect.
The QHE clearly had the potential of elimi-nating in a relatively
straightforward way theproblems of nonuniformity of national
representa-tions of the ohm, their variation in time, and
theirinconsistency with the SI.
To address the problems associated with currentnational
representations of the volt and ohm as dis-cussed above, the CCE at
its 17th meetingestablished through Declaration El (1986),3
"Con-cerning the Josephson effect for maintaining therepresentation
of the volt," the CCE WorkingGroup on the Josephson Effect. The CCE
chargedthe Working Group to propose a new value of theJosephson
frequency-to-voltage quotient consistentwith the SI value based
upon all relevant data thatbecame available by June 15, 1988.
Similarly,recognizing the rapid advances made in under-standing the
QHE since its comparatively recentdiscovery, the CCE established
through Declara-tion E2 (1986),3 "Concerning the quantum Hall
ef-fect for maintaining a representation of the ohm,"the Working
Group on the Quantum Hall Effect.The CCE charged the Working Group
to (i) pro-pose to the CCE, based upon all relevant data thatbecame
available by June 15, 1988, a value of thequantized Hall resistance
consistent with the SIvalue for use in maintaining an accurate and
stablenational representation of the ohm by means of theQHE; and
(ii) develop detailed guidelines for theproper use of the QHE to
realize reliably such arepresentation. 4
Further, the CCE stated its intention to hold its18th meeting in
September 1988 with a view torecommending that both the proposed
new valueof the Josephson frequency-to-voltage quotient andthe
proposed value of the quantized Hall resistancecome into effect on
January 1, 1990. These valueswould be used by all those national
standards
3The complete declaration is given in reference [1], but see
alsoreferences [5] and [7].'The members of the CCE Working Group on
the JosephsonEffect were R. Kaarls, Van Swinden Laboratorium (VSL),
TheNetherlands; B. P. Kibble, National Physical Laboratory(NPL),
U.K.; B. N. Taylor, (NIST); and T. J. Witt, Coordinator(BIPM). The
members of the CCE Working Group on theQuantum Hall Effect were F.
Delahaye (BIPM); T. Endo, Elec-trotechnical Laboratory (lFTL)
Japan; 0, C, Jqones (NPL); Y,Kose, Physikalisch-Technische
Bundesanstalt (PTB), F. R. G.;B. N. Taylor, Coordinator (NIST); and
B. M. Wood, NationalResearch Council of Canada (NRCC), Canada.
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laboratories (and others) that base their representa-tion of the
volt on the Josephson effect, and thatchoose to base their
representation of the ohm onthe QHE. These proposals of the CCE
were subse-quently approved by the CIPM [8] and by the Gen-eral
Conference of Weights and Measures (CGPM)[9] under whose authority
the CIPM functions.
In response to the CCE's directives, each Work-ing Group
prepared a report which focused on thereview and analysis of the
values of the Josephsonfrequency-to-voltage quotient or quantized
Hall re-sistance in SI units that were available by June 15,1988;
and the derivation of a recommended valuefor the purpose of
establishing an accurate and in-ternationally uniform
representation of the volt andof the ohm based on the Josephson
effect and onthe quantum Hall effect, respectively. Submitted tothe
CCE in August 1988, the reports include usefulbackground
information as well as a discussion asto how the new
representations might be used inpractice to express calibration
results. In keepingwith the CCE's charge, the QHE Working Groupalso
prepared a companion report entitled "Techni-cal Guidelines for the
Reliable Measurement of theQuantized Hall Resistance." Because
unbiasedquantized Hall resistance determinations are re-quired for
an accurate and reproducible representa-tion of the ohm based on
the QHE, these guidelinesare of exceptional importance. 5
2. CCE 18th Meeting Discussion andPrincipal Decisions
As an aid to the reader, this section of the reportalso includes
some tutorial information.
2.1 Josephson Effect
2.1.1 Definition of Josephson Constant When aJosephson junction
is irradiated with microwaveradiation of frequency f, its current
vs voltagecurve exhibits steps at highly precise quantizedJosephson
voltages Up. The voltage of the n th stepUj(n), n an integer, is
related to the frequency ofthe radiation by
'The complete reports of the Josephson and Quantum Hall Ef-fect
Working Groups including the "Technical Guidelines"(Rapports BIPM
88/77, 88/8, and 88/9) will appear in the pro-ceedings of the CCE's
18th meeting [2]. Additionally, a com-bined, somewhat condensed
version of the two reports may befound in reference [3] and the
"Technical Guidelines" in refer-ence [10].
U3 (n )=nf/K 3 , (1)
where Kj is commonly termed the Josephson fre-quency-to-voltage
quotient [11]. The WorkingGroup on the Josephson Effect (WGJE)
proposedthat this quotient be referred to as the Josephsonconstant
and, since no symbol had yet beenadopted for it, that it be denoted
by KJ. It followsfrom eq (1) that the Josephson constant is equal
tothe frequency-to-voltage quotient of the n = 1 step.
The theory of the Josephson effect predicts, andthe
experimentally observed universality of eq (1)is consistent with
the prediction, that Kj is equal tothe invariant quotient of
fundamental constants2e/h, where e is the elementary charge and h
is thePlanck constant [11]. For the purpose of includingdata from
measurements of fundamental constantsin the derivation of their
recommended value of Kj,the WGJE assumed that 2e/h =Kp. However, Kj
isnot intended to represent the combination of funda-mental
constants 2e/h.2.1.2 Josephson Effect Reference Standard ofVoltage
The CCE reviewed the report from theWGJE and discussed at some
length the draft rec-ommendation El (1988), "Representation of
thevolt by means of the Josephson effect," preparedjointly by the
WGJE and the Working Group onthe Quantum Hall Effect. The CCE then
agreed:
(i) to use the term "Josephson constant" withsymbol Kj to denote
the Josephson frequency-to-voltage quotient;
(ii) to accept the WGJE's recommended value ofKj, namely,
Kj=(483 597.9±0.2) GHz/V, wherethe 0.2 GHz/V assigned
one-standard-deviationuncertainty corresponds to a relative
uncertainty of0.4 ppm;
(iii) to use this recommended value to define aconventional
value of Kj and to denote it by the
defsymbol Kj_90, so that Kj-90=483 597.9 GHz/V ex-actly. (The
subscript 90 derives from the fact thatthis new conventional value
of the Josephson con-stant is to come into effect starting January
1, 1990,a date reaffirmed by the CCE.) The CCE alsonoted
(iv) that since Kj-90 exceeds the CCE's 1972 con-ventional value
of the Josephson constant by 3.9GHz/V or about 8.065 ppm, the new
representa-tion of the volt will exceed that based on the 1972value
by about 8.065 ppm; and further agreed
(v) that because the purpose of the new volt rep-resentation is
to improve the worldwide uniformityof voltage measurements and
their consistencywith the SI, laboratories which do not base
theirnational representation of the volt on the Joseph-
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The theory of the QHE predicts, and the experi-mentally observed
universality of eq (2) is consis-tent with the prediction, that RK
is equal to theinvariant quotient of fundamental constants h/e
2
[13]. For the purpose of including data from mea-surements of
fundamental constants in the deriva-tion of their recommended value
of RK, theWGQHE assumed that hl/e2 =RK. However, inanalogy with Kj,
RK is not intended to representthe combination of fundamental
constants h/e 2.2.2.2 Quantum Hall Effect Reference Standard
ofResistance The CCE reviewed the report of theWGQHE and discussed
the draft recommendationE2 (1988), "Representation of the ohm by
means ofthe quantum Hall effect," prepared jointly by thetwo
Working Groups. Because of the similaritiesbetween the QHE and the
Josephson effect, thereview and discussion proceeded expeditiously.
In-deed, the second half of point (iii) as given here insection
2.1.2 on the Josephson effect and all ofpoints (v), (vi), and (vii)
were viewed by the CCEas applying to the quantum Hall effect as
well.Also in analogy with the Josephson effect, theCCE agreed:
(i) to use the term "von Klitzing constant" withsymbol RK to
denote the Hall voltage to currentquotient or resistance of the i=
1 plateau;
(ii) to accept the WGQHE's recommended valueof RK, namely,
RK=(25 812.807±0.005) fl, wherethe 0.005 fl assigned
one-standard-deviation uncer-tainty corresponds to a relative
uncertainty of 0.2ppm; and
(iii) to use this recommended value to define aconventional
value of RK and to denote it by the
def'symbol RK-90, so that RK-90o=25 812.807 fl exactly.
The same procedure was followed for draft rec-ommendation E2
(1988) as for El (1988) regardingthe Josephson effect. The final
CIPM English lan-guage version is as follows:
Representation of the Ohm by Means of theQuantum Hall Effect
Recommendation 2 (CI-1988)
The Comit6 International des Poids et Mesures,acting in
accordance with instructions given in
Resolution 6 of the 18th Conf6rence Generale desPoids et Mesures
concerning the forthcoming ad-justment of the representations of
the volt and theohm,
considering-that most existing laboratory reference stan-
dards of resistance change significantly with time,-that a
laboratory reference standard of resis-
tance based on the quantum Hall effect would bestable and
reproducible,
-that a detailed study of the results of the mostrecent
determinations leads to a value of 25 812.807fl for the von
Klitzing constant, RK, that is to say,for the quotient of the Hall
potential difference di-vided by current corresponding to the
plateau i = 1in the quantum Hall effect,
-that the quantum Hall effect, together withthis value of RK,
can be used to establish a refer-ence standard of resistance having
a one-standard-deviation uncertainty with respect to the
ohmestimated to be 2 parts in 107, and a reproducibilitywhich is
significantly better,
recommends-that 25 812.807 fl exactly be adopted as a con-
ventional value, denoted by RK-90, for the von Kl-itzing
constant, RK,
-that this value be used from 1st January 1990,and not before,
by all laboratories which base theirmeasurements of resistance on
the quantum Halleffect,
-that from this same date all other laboratoriesadjust the value
of their laboratory reference stan-dards to agree with RK-90,
-that in the use of the quantum Hall effect toestablish a
laboratory reference standard of resis-tance, laboratories follow
the most recent editionof the "Technical Guidelines for Reliable
Measure-ments of the Quantized Hall Resistance" drawn upby the
Comit6 Consultatif d'tlectricit6 and pub-lished by the Bureau
International des Poids etMesures,
and is of the opinion-that no change in this recommended value
of
the von Klitzing constant will be necessary in theforeseeable
future.
2.3 Practical Implementation of Recommendations
As implied by the discussion of section 1, theresults of voltage
and resistance measurements ex-pressed in terms of representations
of the volt andohm based on the Josephson and quantum Hall
ef-fects, respectively, will have a higher precisionthan the same
measurement results expressed interms of the volt and ohm
themselves. Indeed, thisis one of the principal reasons for
establishing such
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son effect should, on January 1, 1990, adjust thevalue of their
national volt representation so that itis consistent with the new
representation. Further,this consistency should be maintained by
having atransportable voltage standard periodically cali-brated by
a laboratory that does base its representa-tion of the volt on the
Josephson effect;
(vi) that even if future, more accurate measure-ments of Kj
indicate that the recommended valuediffers from the SI value by
some small amount,the conventional value Kj-90 should not be
altered.Rather, the CCE could simply note the differencebetween a
representation of the volt based on Kj-90and the volt; and
(vii) that because an accurate representation ofthe volt is
important to science, commerce, and in-dustry, laboratories should
continue their efforts torealize the volt with greater accuracy,
either di-rectly or indirectly via measurements of fundamen-tal
constants. This could lead to a significantreduction in the
uncertainty assigned to the newvolt representation.
Having concurred on these points, the CCE ed-ited the draft
recommendation El (1988) to bring itto final form. The following
week it was submittedto the CIPM for approval at its 77th meeting
heldon October 4-6, 1988, at the BIPM. After someminor editorial
changes, the CIPM adopted it as itsown recommendation [12]. The
following is theEnglish language version (the French
languageversion is the official one and is given in references[2]
and [12]):
Representation of the Volt by Means of theJosephson Effect
Recommendation 1 (CI-1988)
The Comit6 International des Poids et Mesures,acting in
accordance with instructions given in
Resolution 6 of the 18th Conference Gen6rale desPoids et Mesures
concerning the forthcoming ad-justment of the representations of
the volt and theohm,
considering-that a detailed study of the results of the most
recent determinations leads to a value of 483 597.9GHz/V for the
Josephson constant, Kj, that is tosay, for the quotient of
frequency divided by thepotential difference corresponding to the n
= 1 stepin the Josephson effect,
-that the Josephson effect together with thisvalue of K3 can be
used to establish a referencestandard of electromotive force having
a one-stan-dard-deviation uncertainty with respect to the volt
estimated to be 4 parts in 10', and a reproducibilitywhich is
significantly better,
recommends-that 483 597.9 GHz/V exactly be adopted as a
conventional value, denoted by Kj_90, for theJosephson constant,
K3 ,
-that this new value be used from 1st January1990, and not
before, to replace the values cur-rently in use,
-that this new value be used from this samedate by all
laboratories which base their measure-ments of electromotive force
on the Josephson ef-fect, and
-that from this same date all other laboratoriesadjust the value
of their laboratory reference stan-dards to agree with the new
adopted value,
is of the opinion-that no change in this recommended value
of
the Josephson constant will be necessary in theforeseeable
future, and
draws the attention of laboratories to the fact thatthe new
value is greater by 3.9 GHz/V, or about 8parts in 106, than the
value given in 1972 by theComit6 Consultatif d'Electricit6 in its
DeclarationE-72.
2.2 Quantum Hall Effect
2.2.1 Definition of the von Klitzing Constant TheQHE is
characteristic of certain high mobilitysemiconductor devices of
standard Hall-bar ge-ometry when in a large applied magnetic field
andcooled to a temperature of about one kelvin. For afixed current
I through a QHE device there areregions in the curve of Hall
voltage vs gatevoltage, or of Hall voltage vs magnetic field
de-pending upon the device, where the Hall voltageUH remains
constant as the gate voltage or mag-netic field is varied. These
regions of constant Hallvoltage are termed Hall plateaus. Under the
properexperimental conditions, the Hall resistance of theith
plateau RH(i), defined as the quotient of theHall voltage of the
ith plateau to the current I, isgiven by
RH(i)= UH(i)/I=RK/i, (2)
where i is an integer [13]. Because RH(i) is oftenreferred to as
the quantized Hall resistance regard-less of plateau number, the
Working Group on theQuantum Hall Effect (WGQHE) proposed that
toavoid confusion, the symbol RK be used as the
Hallvoltage-to-current quotient or resistance of thei= 1 plateau
and that it be termed the von Klitzingconstant after the discoverer
of the QHE. It thusfollows from eq (2) that RK=RH(l).
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representations. The question arises, however, asto how such
measurement results should be re-ported in practice. The Working
Groups recog-nized that the potential for significant
confusioninternationally could best be eliminated by havingeach
national standards laboratory adopt the sameapproach. To this end,
in their reports the WorkingGroups identified and considered the
advantagesand disadvantages of three different approaches tothe
reporting problem, two of which are both rig-orous and correct [2].
In the first, new "practicalunits" "V90" and "fl9o" are defined; in
the second,new, so-called "conventional physical quantities"for
electromotive force (and electric potential dif-ference) and
resistance, "Ego" and "R9o0," are de-fined.
The CCE discussed at length the three ap-proaches identified by
the Working Groups andconcluded that there was an alternative
solution,similar to the Working Groups' third approach,that is also
rigorous but avoids
(i) defining new practical units of emf and resis-tance that are
likely to differ from the volt and ohmby small amounts and which
would be parallel toand thus in competition with the volt and
ohm.(Defining such units automatically leads to practi-cal
electrical units for current, power, capacitance,etc., thereby
giving the appearance that a completenew system of electrical units
has been establishedoutside of the SI.) The CCE's alternative
solutionalso avoids
(ii) defining new conventional physical quantitiesfor emf and
resistance which are likely to differfrom traditional or true emf
and resistance by smallamounts. (Defining such quantities
automaticallyleads to conventional physical quantities for
cur-rent, power, capacitance, etc.; and to the peculiarsituation
of, for example, the same standard cellhaving both a conventional
emf and a true emf.)Further, the alternative solution avoids
(iii) the use of subscripts or other distinguishingsymbols of
any sort on either unit symbols or quan-tity symbols. (With the
elimination of such sub-scripts and symbols, for example, those
denotingparticular laboratories or dates, the national stan-dards
laboratories can avoid giving the impression
6 As noted by the CCE [2], the Josephson and quantum Halleffects
and the values Ki-90 and RK-90 cannot be used to definethe volt and
ohm. To do so would require a change in the statusof the
permeability of vacuum p0 from an exactly defined con-stant,
thereby abrogating the definition of the ampere. It wouldalso give
rise to electrical units which would be incompatiblewith the
definition of the kilogram and units derived from it.
to the users of their calibration services that thereis more
than one representation of the volt and ofthe ohm in general use,
that there may be signifi-cant differences among national
realizations of thenew volt and ohm representations, and that
eitherthe national realizations or the new representationsdiffer
significantly from the SI.)
The CCE's solution, which was affirmed by theCIPM at its 77th
meeting [12] and which all na-tional standards laboratories are
requested to fol-low, is indicated in the following variation of
theexample given by the CCE [2] (the treatment ofresistance
measurements is strictly analogous):
The emf E of an unknown standard cell cali-brated in terms of a
representation of the volt basedon the Josephson effect and the
conventional valueof the Josephson constant Kj_90, may be
rigorouslyexpressed in terms of the (SI) volt V as (to
bespecific):
E=(1.018 123 45) V+e, (3)
where E represents the total uncertainty, in volts,and is
composed of the following two components:AE, the combined
uncertainty associated with thecalibration itself and with the
realization of theJosephson effect volt representation at the
particu-lar standards laboratory performing the calibration;and AA,
the uncertainty with which the ratioKj3 9 0 /KJ is known (i.e., it
is assumed that Kj 90 /KJ= 1±+-A4). According to Recommendation
1(CI-1988), A4 is 4 parts in 107 or 0.4 ppm (assignedone standard
deviation).
Since, by international agreement, A4 is com-mon to all
laboratories, the two uncertainties AEand AA need not be formally
combined to obtainthe total uncertainty E but may be separately
indi-cated. Hence, the measured emf E may be ex-pressed as
E=(1.018 123 45) V±AE (4)
for all practical purposes of precision electricalmetrology and
trade, with £4 appearing separatelyon the calibration certificate
when the precision ofthe calibration warrants it. If, for example,
AEIE issignificantly greater than 0.4 ppm, AA may beomitted with
negligible effect.
An example of the wording that might be usedon a NIST Report of
Calibration for a standard cellenclosure for the case where £A may
not be omit-ted and which is a variation of the wording givenin an
example developed by the CCE [2], is as fol-lows:
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Sample Hypothetical NIST Calibration Report
This standard cell enclosure was received (date) under power at
itsnormal operating temperature.
The values given in the table below are based on the results of
dailymeasurements of the differences between the emfs of the cells
in thisstandard and those of NIST working standards calibrated in
terms ofthe Josephson effect using the new conventional value of
the Josephsonconstant internationally adopted for use starting
January 1, 1990 (seeNote A). The measurements were made in the
period from (date) to(date).
Cell emf Uncertaintynumber (volts, V) (microvolts, XV)
1 1.018 119 85 0.272 1.018 133 77 0.273 1.018 126 42 0.274 1.018
141 53 0.27
(Information relating to the measurements and their
uncertainties to begiven here.)
Note A
The value of the Josephson constant used in this calibration,
namely,Kj- 90 =483 597.9 GHz/V exactly, is that adopted by
international agree-ment for implementation starting on January 1,
1990, by all nationalstandards laboratories that base their
national representation of the volt(i.e., their national "practical
unit" of voltage) on the Josephson effect.Since all such
laboratories now use the same conventional value of theJosephson
constant while prior to this date several different values werein
use, the significant differences which previously existed among
thevalues of some national representations of the volt no longer
exist.Moreover, the national standards laboratories of those
countries that donot use the Josephson effect for this purpose are
requested to maintaintheir own national representation of the volt
so as to be consistent withthe above conventional value of the
Josephson constant, for example,through periodic comparisons with a
laboratory that does use theJosephson effect. An ideal
representation of the volt based on theJosephson effect and Kj- 90
is expected to be consistent with the volt asdefined in the
International System of Units (SI) to within an assignedrelative
one-standard-deviation uncertainty of 0.4 ppm (0.41 ,uV for anemf
of 1.018 V). Because this uncertainty is the same for all
nationalstandards laboratories, it has not been formally included
in the uncer-tainties given in the table. However, its existence
must be taken intoaccount when the utmost consistency between
electrical and nonelectri-cal measurements of the same physical
quantity is required.
2.4 Future Work on Electrical Units Hall effect, led the CCE to
adopt the followingformal recommendation which was also approvedThe
ideas agreed upon by the CCE as given in by the CIPM at its 77th
meeting [12].
point (vii) in Sect. 2.1.2 on the Josephson effect,and which
apply equally as well to the quantum
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Realization of the Electrical SI Units
Recommendation E3 (1988)
The Comit6 Consultatif d'Electricit6recognizing-the importance
to science, commerce and in-
dustry of accuracy in electrical measurements,-the fact that
this accuracy depends on the ac-
curacy of the reference standards of the electricalunits,
-the very close ties that now exist betweenelectrical metrology
and fundamental physical con-stants,
-the possibility of obtaining more accurate ref-erence standards
of the electrical units either di-rectly from the realizations of
their definitions orindirectly from measurements of fundamental
con-stants, and
-the continuing need to compare among them-selves independent
realizations of the units and in-dependent measurements of
fundamental constantsto verify their accuracy,
recommends-that laboratories continue their work on the
electrical units by undertaking direct realizations ofthese
units and measurements of the fundamentalconstants, and
-that laboratories pursue the improvement ofthe means for the
international comparison of na-tional standards of electromotive
force and electri-cal resistance.
3. Conclusion
The apparatus currently being used by the na-tional standards
laboratories is such that the totalexperimental uncertainty
associated with a particu-lar national representation of the volt
based on theJosephson effect generally lies in the range 0.01 to0.2
ppm. As a consequence, with the worldwideadoption starting January
1, 1990, of the new con-ventional value of the Josephson constant
KJ_90, allnational representations of the volt should beequivalent
to within a few tenths of a ppm. Simi-larly, the total experimental
uncertainty associatedwith the measurement of quantized Hall
resistancesalso generally lies in the range 0.01 to 0.2 ppm.Hence,
with the worldwide adoption starting onJanuary 1, 1990, of a new
representation of theohm based on the QHE and the conventional
valueof the von Klitzing constant RK-90, all national
rep-resentations of the ohm should also be equivalent
to within a few tenths of a ppm. Moreover, thesenew national
volt and ohm representations shouldbe consistent with the volt and
the ohm to betterthan 0.5 ppm.
In the U.S., the value of the present national rep-resentation
of the volt maintained by NIST willneed to be increased on January
1, 1990, by about9.26 ppm to bring it into agreement with the
newrepresentation of the volt. This is sufficiently largethat
literally thousands of electrical standards,measuring instruments,
and electronic systemsthroughout the Nation will have to be
adjusted orrecalibrated in order to conform with the new
rep-resentation. Most other countries will be requiredto make a
similar change in the value of theirpresent representation of the
volt as can be seenfrom figure 1. On the same date, the value of
theU.S. representation of the ohm maintained byNIST will need to be
increased by about 1.69 ppmto bring it into agreement with the new
representa-tion of the ohm based on the quantum Hall effect.This
too is an amount which is of significance tomany existing
standards, instruments, and systems.
1
0
-1
-2
-3
-4E& -5
-6
-7
-8
-9
-10
NEW VOLTREPRESEN-TATION
U.S.S.R.
0 ppm
-3.565 ppm
+4.500 ppm
_~ . x~x~x~xzFRANCE+1.323 ppm
-1 ALL OTHERCOUNTRIES_ 1 -1.199 ppm~~~~~~~U.S.
-6.741 ppm
-8.065 ppm
-9.264 ppm
Figure 1. Graphical comparison of the value of the present
rep-resentation of the volt of various countries as based on
theJosephson effect, with the new representation of the volt
basedon the Josephson effect and the CIPM conventional value of
theJosephson constant K 3-go which is to come into effect
startingon January 1, 1990. The value of the volt representation
indi-cated by "All Other Countries" is based on the
conventionalvalue of the Josephson constant stated by the CCE in
1972,namely, 483 594 GHz/V. The countries that currently use
thisvalue include Australia, Canada, Finland, F.R.G., G.D.R.,Italy,
Japan, The Netherlands, and the U.K. The BIPM uses this
value as well, but NIST uses 483 593.420 Gflz/V. Thus, as
thefigure shows, on January 1, 1990, the value of the present
U.S.volt representation will need to be increased by 9.264 ppm
tobring it into conformity with the new representation.
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The change required in the value of the nationalrepresentation
of the ohm of other countries variesbetween a decrease of a few
tenths of a ppm to anincrease in excess of 3 ppm.
Since A=V/A where A is the ampere as definedin the SI; and W=V
2/fl where W is the watt asdefined in the SI, the 9.264 ppm and
1.69 ppm in-crease in the U.S. representation of the volt and ofthe
ohm, respectively, imply that on January 1,1990, (i) the U.S.
representation of the ampere willincrease by about 7.57 ppm and
(ii) the U.S. electri-cal representation of the watt will increase
byabout 16.84 ppm. Because an ideal volt representa-tion based on
the Josephson effect and Kj- 90 is ex-pected to be consistent with
the volt to within anassigned relative one-standard-deviation
uncer-tainty of 0.4 ppm; and an ideal ohm representationbased on
the QHE and RK-90 is expected to be con-sistent with the ohm to
within an assigned one-stan-dard-deviation uncertainty of 0.2 ppm,
ampere andwatt representations derived from such ideal voltand ohm
representations via the above equationsare expected to be
consistent with the ampere andwatt to within a
one-standard-deviation uncer-tainty of 0.45 ppm and 0.83 ppm,
respectively.
The CCE strongly believes, and the author fullyconcurs, that the
significant improvement in the in-ternational uniformity of
electrical measurementsand their consistency with the SI which will
resultfrom implementing the new representations of thevolt and ohm
will be of major benefit to science,commerce, and industry
throughout the world; andthat the costs associated with
implementing thenew representations will be far outweighed bythese
benefits.
[6] v. Klitzing, K., Dorda, G., and Pepper, M., Phys. Rev.Lett.
45, 494 (1980).
[7] BIPM Com. Cons. Electricit6 17, E6, E88; E10, E92(1986).
[8] Reference [4], p. 10.[9] BIPM Comptes Rendus 18e Conf. G6n.
Poids et Mesures,
p. 100 (1987). See also Giacomo, P., Metrologia 25,
113(1988).
[10] Delahaye, F., Metrologia 26, No. 1 (1989), to be
published.[11] Clarke, J., Am. J. Phys. 38, 1071 (1970).[12] BIPM
Proc.-Verb. Com. Int. Poids et Mesures 56 (1988),
to be published.[13] The Quantum Hall Effect, eds. R. E. Prange
and S. M.
Girvin, Springer-Verlag, NY (1987). This book provides
acomprehensive review of the quantum Hall effect.
About the author: Barry N. Taylor, a physicist, ishead of the
Fundamental Constants Data Center inthe NIST National Measurement
Laboratory andChief Editor of the Journal of Research of the
Na-tional Institute of Standards and Technology.
4. References
[1] Taylor, B. N., J. Res. Natl. Bur. Stand. 92, 55 (1987).[2]
BIPM Com. Cons. tlectricit6 18 (1988), to be published.[3] Taylor,
B. N., and Witt, T. J., Metrologia 26, No. 1 (1989),
to be published.[4] BIPM Proc.-Verb. Com. Int. Poids et Mesures
54, 35
(1986). See also Kaarls, R., BIPM Proc. Verb. Int. Poids
etMesures 49, Al, 26 (1981); and Giacomo, P., Metrologia18, 41
(1982).
[5] Giacomo, P., Metrologia 24, 45 (1987).
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
A Supercritical Fluid Chromatographfor Physicochemical
Studies
Volume 94 Number 2 March-April 1989
Thomas J. Bruno A supercritical fluid chromatograph has
instrument has recently been applied tobeen designed and
constructed to make the measurement of diffusion coeffi-
National Institute of Standards physicochemical measurements,
while cients of toluene in supercritical carbonretaining the
capability to perform dioxide at a temperature of 313 K, and
and Technology, chemical analysis. The physicochemical pressures
from 133 to 304 barBoulder, CO 80303 measurements include diffusion
coeffi- (13.3-30.4 MPa). The data are dis-
cients, capacity ratios, partition coeffi- cussed and compared
with previouscients, partial molar volumes, virial measurements on
similar systems.coefficients, solubilities, and molecularweight
distributions of polymers. In this Key words: diffusion
coefficients; super-paper, the apparatus will be described in
critical fluid chromatography.detail, with particular attention
given toits unique features and capabilities. The Accepted:
December 11, 1988
1. Introduction
The methods of chromatography have been ap-plied to
physicochemical problems such as thermo-physical property
determination for nearly 30 years[1,2]. This application of
chromatography to non-analytical problems stems from an
understanding ofthe physical and chemical processes which areknown
to occur during chromatographic separa-tions [3]. As an example,
since hydrogen bondingcan play a role in chromatographic
separation, wemay apply chromatography to the study of hydro-gen
bonding thermodynamics [4].
The development of chromatography during thelast 80 years can be
divided into distinct historicalperiods [5], each with its
innovations, fads and fail-ures. During the current period, we have
seen theemergence of supercritical fluid chromatography(SFC, in
which the carrier is a fluid held above itscritical point). Some
properties of a typical super-critical fluid can be seen in table
1. The density of
Table 1. Comparison of representative fluid properties
Gas Liquid Supercritical fluid
Density, p 10-3 1 0.7g/mL
Diffusivity, D 10- 5X 10-6 lo-3cm
2/s
Dynamic viscosity, q 10-4 10-2 10-4g/(cm.s)
the supercritical fluid is very similar to that of aliquid
phase. This property explains the greatly en-hanced solvation power
of the supercritical fluidwith respect to the gas phase. The
viscosity of thesupercritical phase closely resembles that of the
gasphase, thus allowing for easy mass transfer. Thethermal
conductivity (though not shown in table 1)
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is also relatively large, as one would expect fromdensity and
viscosity considerations. The diffusiv-ity (self-diffusion) of the
supercritical phase is inter-mediate between that of a gas and a
liquid. Thisproperty gives the supercritical fluid the
advantageover liquid-liquid extraction. There are many ex-cellent
reviews describing the advantages and ap-plications of SFC, and the
reader is referred tothese for additional details [6,7].
Most of the applications of SFC found in theliterature involve
analytical or separation prob-lems. The application to
physicochemical studieshas been relatively slow due to many
experimentaldifficulties. Nonetheless, SFC has been applied to
anumber of thermophysical problems by severalgroups. This has
included the measurement of ca-pacity ratios, partition
coefficients, binary diffusioncoefficients, partial molar volumes,
virial coeffi-cients, solubilities, and polymer molecular
weightdistributions. In this work, the physicochemical
su-percritical fluid chromatograph has been applied tothe
measurement of binary diffusion coefficients[8-16]. This diffusion
coefficient describes the ten-dency of the solute to diffuse into
the carrier (usu-ally referred to as the solvent).
quantity in chromatography, since it describes theefficiency of
the chromatographic system [24].This quantity is the width of a
peak (as designatedby its variance, a-2, in length units as opposed
totime units) relative to the distance traversed insidethe column
or tube (i.e., the length of the tube, L):
H=&2/L. (2)
In a straight tube, the concentration profile of thesolute in
the carrier will become Gaussian-likewhen H
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FluidReservoirs
FlameIonizationDetector
Injector legral(r/COml)tIterOven
Figure 1. Schematic diagram showing the major components of the
supercritical fluid chromatograph used in this work.
as carbon dioxide [27,28]. Before the carrier fluidenters the
pump head, it is first passed through aheat exchanger which
consists of a 600-cm lengthof stainless steel (304) tubing
(0.076-cm inside di-ameter, 0.16-cm outside diameter). This heat
ex-changer insures that fluid is delivered to the pumphead as a
liquid. The vortex tube is operated in anintermediate mode (between
maximum cooling andmaximum temperature difference, with an
appliedair pressure of 0.7 MPa), and provides a carrierstream
temperature of -20 'C. This minimizes va-por locking and cavitation
inside the pump heads.
The pump is followed by a pulse dampener andpressure transducer.
The pulse dampener is a coilof flattened stainless steel tubing
(0.64-cm outsidediameter) which absorbs much of the low level
pul-sation not handled by the electronic compensation.This
component is necessary since no piston pumpcan operate in true
pulse-free fashion. The pressuretransducer is a strain gauge device
which is cali-brated periodically using a high precision
Bourdontube transfer standard. This transfer standard is it-self
calibrated using a dead weight pressure balancetraceable to the
NIST primary standard. The un-certainty in the measured pressure
has been deter-mined at ±0.40 bar (±0.040 MPa).
After leaving the pressure transducer the fluidenters a heat
exchanger (fig. 2) inside the columnoven, where the flash to
supercritical temperatureoccurs. This heat exchanger is a 300-cm
section ofstainless steel (304) tubing (0.32-cm outside diame-ter,
0.07-cm inside diameter). A vibrating tube den-simeter is
downstream from the heat exchanger, toallow independent density
measurements of thecarrier if desired. The densimeter places
operatingconstraints upon the entire system (41 MPa maxi-mum
pressure, 160 'C maximum temperature), andmust be removed for
higher temperature or pres-sure operation.
Following the solvent delivery system is thesample injector. The
injector used in this work isthe flow-through extractor coupled
with a high-pressure chromatographic sampling valve shownin figure
3 [29]. This arrangement is most satisfac-tory for experiments
involving repetitive injectionsof the same solute. An aliquot of
solvent-bornesample is syringe-deposited into the extractor.
Theextractor is then heated and evacuated to removethe sample
solvent. After the solvent is removed(an operation which takes
approximately 5 min-utes), the extractor is filled with the
supercriticalcarrier (solvent) at the same temperature and
pres-
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ThermalTempering Block
Aluminum'Racetrack'
DiffusionTube
inertia Block
-5 Heater; ( ~ Block
Vibrating TubeDensimeter
Shimming Heater(one of three)
Figure 2. Diagram of the thermostatted column area containing
the diffusion tube and vibrating tube densimeter.
Sample Loop -' VentValveTo Column
VacuumValve
6-portSample Valve
l ~~~~~Carrier Inl
FromCarrier ExrcoSource
Air Bath
Figure 3. Schematic diagram of the sample injection system.
sure as that in the column. The sample is thenloaded into the
sample loop to allow injection intothe carrier stream. The
injection is achieved auto-matically under computer control using
an pneu-matic actuator (with helium as the working fluid)equipped
with pilot valves and a large ballast vol-ume to provide fast
switching.
Upon injection, the carrier-borne solute is trans-ported to the
column area (fig. 2) inside of a modi-fied commercial forced-air
oven capable ofmaintaining a temperature of 350 'C. The
majormodifications to the oven include baffles whichpromote uniform
air movement and an inert gaspurge line for safety. The column area
contains theheart of the physicochemical experiment, whichmay
consist of a coated or uncoated capillary or apacked column. In the
case of diffusion coefficientmeasurements, the column consists of a
long un-coated capillary. This capillary is currently a 3040-cm
long continuous section of 316 stainless steel
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
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tubing (0.159-cm outside diameter, 0.025+±0.0013-cm inside
diameter) which is connected directly tothe injection valve. The
length of the diffusion tubewas determined by weight, using a
calibrationequation. This tube is coiled in a 30-cm diameter,held
inside of an aluminum racetrack which inte-grates out local
temperature variations. The race-track also serves as the support
for the vibratingtube densimeter referred to earlier.
Temperaturemeasurement is provided by a platinum resistanceprobe
located in the center of the racetrack. Sixpairs of
gradient-sensing thermocouples (type-j,thermally tempered)
referenced to the main ther-mometer provide an indication of
temperaturenonuniformity. This nonuniformity is then reducedto a
negligible level using a set of manually con-trolled low power
shimming heaters. The race-track is supported inside the oven by
titanium rodsof 0.64-cm diameter. The low thermal conductivityof
the titanium limits heat transfer into or out of theoven. The
temperature uniformity of the racetrackhas been measured at ±0.015
'C at temperatures ator above 40 'C.
After leaving the racetrack, the solute is trans-ported to the
detector (fig. 4) by the diffusion tube,as shown in figure 2. The
detector is a modifiedchromatographic flame ionization detector
con-tained in a separate oven directly above the columnoven. The
major modifications made to the com-mercial unit were in the sample
and jet gas inletlines. Provisions have also been made for the
intro-duction of a make-up gas where needed. A fused
IgnitorInsulator
Signal Collector Probe
Flame InsulatrFlame Jet os-Insulator Polarizing Electrode
Restrictor
.... t- Air Diffuser
Detector Heater AirHydrogen
Makeup Ca,
l > ~~PTFE GlandFrom Diffusion Tube
Figure 4. Schematic diagram of the modified flame
ionizationdetector.
quartz capillary restrictor, attached to the diffusiontube,
releases the solute and carrier directly intothe flame. The
detector is operated to provide anapproximate sensitivity of 10ll
mol/s. The insidediameter and length of this restrictor capillary
ischosen so as to maintain the carrier velocity be-tween 2 and 6
cm/s. The temperature of the detec-tor is maintained at 300 'C to
prevent the carrierfrom cooling (and possibly solidifying) upon
de-compression. This cooling action has been noted asa cause of
baseline "spiking" in unheated detectors.The output from the
detector is logged on a com-mercial electronic integrator, from
which the re-tention times and peak widths may be extracted.
4. Results and Discussion
The main sources of error in this experimentstem from the
diffusion tube itself, the sample injec-tion process, pressure drop
across the tube, and ad-sorption of the solute on the tube walls. A
briefdiscussion of these errors will be represented here,especially
with respect to this apparatus. A numberof excellent general
reviews are available in the lit-erature [30-34].
The theory of Taylor and Aris, summarized ear-lier, is strictly
based on straight tubes of circularcross-section and uniform inside
diameter. Forpractical reasons the tube is held inside the oven asa
helical coil rather than a straight length of tubing.In addition,
tubes of perfectly circular cross-sec-tion and uniform inside
diameter are an idealizationnot available in the laboratory. We
must thereforedetermine to what extent our experimental appara-tus
departs from the theoretically assumed condi-tions. It can be shown
[34] that the coiling of thediffusion tube will have no harmful
effects if cer-tain criteria are met. The ratio (ct) of the radius
ofthe coil to that of the inside diameter of the diffu-sion tube
should be greater than 100. In the presentapparatus, the radius
ratio is approximately 1200.Another requirement is that the
inequalityDe2Sc
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Volume 94, Number 2, March-April 1989
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(for example, a sinusoidal variation superimposedon the radius
has been considered [34]). Selectionof a good quality seamless
tubing will help mini-mize most of the problems associated with
tubenonuniformity. It should be noted that the uncer-tainty in the
tube's inside diameter cited in theexperimental section (-5
percent) is a worst caselimit, but this value has been used in the
overallerror analysis. Since the tube length and radius areboth
temperature dependent, corrections are ap-plied to account for the
effect of thermal expansionon these parameters. The effect of the
applied pres-sure on the internal tube radius is negligible.
Theeffects of connections between the tubes have beenminimized by
design (only one such connection isused, having the same internal
radius as the diffu-sion tube), and are considered to be
negligible.
The errors associated with injection have beenaddressed in
several ways. The sample is intro-duced at the same pressure and
temperature as thatof the carrier stream, and at infinite dilution.
Thevolume of the sample loop is very small (= 3 .0 pAL)as compared
with the volume of the diffusion tube.The sampling loop is switched
into the carrierstream only for a short time, and then switchedback
to the fill position. This has been found todecrease problems from
solute adsorption. The in-jection process is done extremely fast
using the pi-lot valve system, resulting in a negligible
pressurepulse.
The pressure drop across the diffusion tube hasbeen measured at
between 0.3 and 0.6 bar (0.03-0.06 MPa). Since this is on the order
of- the experi-mental error of the pressure measurement,
nocorrections are made, but close proximity of thecritical point is
avoided. Measurements madewithin a few degrees of the critical
temperatureand near the critical pressure will require
consider-ation of this density gradient effect. Solute adsorp-tion
on the inside walls of the tube has not been aproblem in the
present work, as judged from thepeak symmetry. Adsorption is often
a problemwhen using highly polar solutes at relatively lowcarrier
density and temperature. To address this is-sue, diffusion tubes
which have larger internal radiiare used so as to decrease the
surface area-to-vol-ume ratio. This must be done with consideration
ofthe trade-off in radius ratio co, and the secondaryflow effects
which may result.
As an example of the operation of this apparatus,measurements of
binary diffusion coefficients oftoluene (at infinite dilution) in
supercritical carbondioxide are presented in table 2 [35]. The
measure-ments were made at 313.83±0.02 K, and at pres-
sures from 133 to 304 bar (13.3 to 30.4 MPa). Thecarrier fluid
density corresponding to these tem-perature-pressure pairs was
calculated from the32-term Benedict-Webb-Rubin (BRW) equation
ofstate for carbon dioxide [36], and ranged from0.746 to 0.910
g/cm3 . As discussed earlier, the rawchromatographic data obtained
were the peakwidths and breakthrough times (the term "reten-tion
time" being considered inappropriate due tothe absence of a
stationary phase). At each density,15 separate determinations were
made, furnishingthe experimental uncertainty for the error
analysis.The combined uncertainties of all measured quanti-ties
provide an overall estimate of between 5 and 6percent for the data
presented here. The diffusioncoefficients are all on the order of
10-4 cm 2 /s, anddecrease with increasing carrier density as can
beseen in figure 5. These data fit in quite well withprevious data
on lower molecular weight aromat-ics, although most of the previous
data were takenat somewhat lower densities [8-11,14]. Compari-sons
of this data with several predictive approachesare as yet
incomplete and will be presented in thefuture. Current work also
includes a study of ahomologous series of straight chain
hydrocarbons,and several members of the carotene family.
Table 2. Measured binary diffusion coefficients, D12 , of
toluenein supercritical carbon dioxide, at 313.83±0.02 K
CO2 density D 12X 104(g/mL) cm2/s
0.746 1.300.801 1.230.893 1.210.867 1.200.890 1.190.910 1.19
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Journal of Research ofVolume 94, Number 2, March-April 1989
the National Institute of Standards and Technology
0.70 0.725 0.750 0.775 0.800 0.825 0.850 0.875 0.900 0.925/p
(g/cm3)
Figure 5. Plot of the binary diffusion coefficient, D12 , of
toluene in supercriticalcarbon dioxide at 313.83 K versus the
density of supercritical carbon dioxide.
5. Acknowledgment
The financial support of the United States De-partment of
Energy, Office of Basic Energy Sci-ences, Division of Engineering
and Geosciences, isgratefully acknowledged.
About the author: Thomas J. Bruno is a physicalchemist in the
Thermophysics Division of the NationalEngineering Laboratory at the
National Institute ofStandards and Technology, Boulder, CO.
6. References
[1] Conder, J. R., and Young, C. L., Physicochemical
mea-surement by gas chromatography (John Wiley and Sons,Chichester,
1979).
[2] Laub, R. J., and Pecsok, R. L., Physicochemical
applica-tions of gas chromatography (John Wiley and Sons, NewYork,
1978).
[3] Bruno, T. J., Proc. Second Symp. Energy Eng. Sci.,Argonne
Natl. Lab., CONF-8404123, (1984) pp. 78-85.
[4] Bruno, T. J., and Martire, D. E., J. Phys. Chem. 87,
2430(1983).
[5] Bruno, T. J., Proc. Fifth Symp. Energy Eng. Sci., Ar-gonne
Natl. Lab., CONF-8706187, (1987) pp. 52-60.
[6] Jackson, W. P., Richter, B. E., Fjeldsted, J. C., Kong,R.
C., and Lee, M. L., Ultrahigh resolution chromatogra-phy, Ahuja,
S., ed., ACS Symposium Series, #250, Wash-ington, D.C. (1984).
[7] White, C. M., and Houck, R. K., J. High Res.
Chromatogr.Chromatogr. Comm. 9, 3 (1986).
[8] Van Wassen, V., Swaid, I., and Schneider, G. M., Angew.Chem.
Int. Ed. Eng. 19, 575 (1980).
[9] Swaid, I., and Schneider, G. M., Ber. Bunsenges. Phys.Chem.
83, 969 (1979).
[10] Wilsch, A., Feist, R., and Schneider, G. M., Fluid
PhaseEquilibria 10, 299 (1983).
[11] Laver, H. H., McManigill, D., and Board, R. D., Anal.Chem.
55, 1370 (1983).
[12] Feist, R., and Schneider, G. M., Sep. Sci. Tech. 17,
261(1982).
[13] Springston, S. R., and Novotny, M., Anal. Chem. 56,
1762(1984).
[14] Sassiat, P. R., Mourier, P., Claude, M. H., and Rossett,R.
H., Anal. Chem. 59, 1164 (1987).
[15] Altares, T., J. Polym. Sci., Part C, Polymer Lett. 8,
761(1970).
[16] Hartmann, W., and Klesper, E., J. Polym. Sci. PolymerLett.
15, 713 (1977).
[17] Giddings, J. C., and Seager, S. L., J. Chem. Phys. 33,
1579(1960).
[18] Wasik, S. P., and McCulloh, K. E., J. Res. Natl. Bur.Stand.
(U.S.) 73A, 207 (1969).
[19] Grushka, E., and Maynard, V., J. Chem. Educ. 49,
565(1972).
[20] Taylor, G., Proc. R. Soc. London 219A, 186 (1953).[21]
Taylor, G., Proc. R. Soc. London 223A, 446 (1954).[22] Taylor, G.,
Proc. R. Soc. London 225A, 473 (1954).[23] Aris, R., Proc. R. Soc.
London 235A, 67 (1956).[24] Grob, R. L., Modern Practice of Gas
Chromatography,
2nd ed., (John Wiley and Sons, New York, 1985).[25] Levenspiel,
O., and Smith, K., Chem. Eng. Sci. 6, 227
(1957).[26] Golay, M. J. E., in Gas chromatography, Desty, D.
H., ed.
(Butterworth, London, 1958).[27] Bruno, T. J., J. Chem. Educ.
64, 987 (1987).[28] Bruno, T. J., Liq. Chromatogr. 4, 134
(1986).
111
1.325
1.300
1.275D,2 x 10'
(cm IS)1.2 50
1.225
1.200
1.1750.950
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
[29] Bruno, T. J., Solvent-free injection in SFC using
sinteredglass deposition, J. Res. Natl. Inst. Stand. Technol.
(U.S.)93, 655 (1988).
[30] Marrero, T. R., and Mason, E., J. Chem. Phys. Ref. Data 1,3
(1972).
[31] Balenovic, Z., Myers, M. N., and Giddings, J. C., J.
Chem.Phys. 52, 915 (1970).
[32] Grushka, E., and Kikta, E. J., J. Phys. Chem. 78,
2297(1974).
[33] Cloete, C. E., Smuts, T. W., and deClerk, K., J.
Chro-matogr. 121, 1 (1976).
[34] Alizadeh, A., Nieto de Castro, C. A., and Wakeham, W.A.,
Int. J. Thermophys. 1, 243 (1980).
[35] Bruno, T. J., Int. J. Thermophys., in press.[36] Ely, J.
F., Proc. 63rd Gas Processors Assn. Annual Conv.
(1984), p. 9.
Appendix 1
Some important hydrodynamic parameters:
1. Radius ratio:co =Rcao
2. Reynoldsnumber:Re=2ao fiop/i1
3. Schmidtnumber:Sc =q/p D12
4. Dean number:De=Re o-112
where Rc is the overallradius of the coil, and ao isthe internal
radius of thediffusion tube. The effectof diffusion tube coilingmay
be considered negli-gible if the radius ratio isgreater than
100.
where ii0 is the averagecarrier fluid velocity, p isthe carrier
density, and 71is the carrier viscosity.The fluid flow is
consid-ered laminar if theReynolds number is lessthan 2000.
where D12 is the binarydiffusion coefficient.
The effects of diffusiontube coiling can be con-sidered
negligible ifSCDe2
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Relation Between Wire Resistance and FluidPressure in the
Transient Hot- Wire Method
Volume 94 Number 2 March-April 1989
H. M. Roder and R. A. Perkins The resistance of metals is a
function of sure, of -2X l0- MPa-' be used to ac-applied pressure,
and this dependence is count for the pressure dependence of
National Institute of Standards large enough to be significant
in the cal- the platinum wire's resistance.and Technology, ibration
of transient hot-wire thermalBoulder, CO 80303 conductivity
instruments. We recom- Key words: fluid; platinum; pressure;
re-
mend that for the highest possible accu- sistance; thermal
conductivity; transientracy, the instrument's hot wires should
t-wire.be calibrated in situ. If this is not possi-ble, we
recommend that a value of -y,the relative resistance change with
pres- Accepted: December 5, 1988
1. Introduction
During the last decade the transient hot-wire hasevolved as the
primary method for the measure-ment of thermal conductivity of
fluids. In these sys-tems, the wire, usually platinum, is used both
as theheating element and as the temperature sensor. Theprimary
variable measured is the change of resis-tance of the wire as a
function of time. The wire isimmersed directly in the fluid, and
any pressureexperienced by the fluid is transmitted to the wire.It
is well known that the resistance of metalschanges with applied
pressure (see, for example,ref. [1]). What is perhaps not widely
appreciated isthat this effect is substantial enough to be
detectedat the relatively low fluid pressures encountered inthe
typical transient hot-wire measurement of ther-mal conductivity. In
this paper we report resis-tance measurements on 12.5 ,tm diameter
platinumwires as a function of pressure up to 70 MPa.
2. Method
In the transient hot-wire method the resistance-temperature
relation of the platinum wires must bedefined accurately to achieve
reliable results forthermal conductivity measurements. In our
ver-sions of this method [2,3] we have opted for a wirecalibration
in situ. In both the low-temperatureversion, 70 to 300 K, [2] and
the high-temperatureversion, 300 to 600 K, [3] we use a
Wheatstonebridge to measure resistances. Compensation forend
effects is provided by placing the long hot wirein one working arm
of the bridge and a shorter,compensating, wire in the other. In
contrast tomost other instruments where times are measuredat a null
voltage point, in our instruments thevoltages developed in the
bridge are measured di-rectly as a function of time with a fast
digital volt-meter.
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Before making each thermal conductivity mea-surement, the bridge
is balanced using a small sup-ply voltage, 50 to 100 mV, to give an
output asnear to zero as possible. We have one calibratedstandard
resistor in each side of the bridge. Thevoltage drops measured
across the standard resis-tors yield the currents in each side of
the bridge.Resistances are then determined in terms of voltagedrops
across the elements of the bridge, that is thehot wires, the leads,
and the adjustable balancingresistors. The hot-wire resistances
measured duringthe balancing of the bridge, together with the
celltemperatures determined from the calibrated plat-inum
resistance thermometer mounted on the cell,are taken as the in situ
calibration of the wires.
As described in reference [2] the resistance rela-tion for each
wire was represented by an analyticalfunction of the type,
R(T,P)=A +BT+CT2+DP, (1)
where R (T,P) is the wire resistance, T is the tem-perature, and
P is the applied pressure. The pres-sure dependence was small, but
statisticallysignificant. In the low-temperature system [2]
thehigh-pressure cell closure can accommodate onlythree leads.
These leads are the two current leadsand one potential tap at the
corner of the bridge.Additional potential taps are placed outside
thehigh-pressure cell. Since there are still short sec-tions of the
leads within the cell we cannot mea-sure the voltage drops across
each hot wiredirectly. The leads are steel and copper, and
wereaccounted for by using resistance tables and bymeasuring the
length and diameter of each piece.For the low-temperature system
[2] we could not
High PressureCell Boundary
i Long'Hot Wire E I
Power- _ lSupply c ,
be certain that the observed pressure depedence re-sulted only
from the platinum wire since other ex-planations were also
possible.
3. Apparatus
During the last three years we have modifiedand improved the
low-temperature system to en-able us to measure the thermal
diffusivity of thefluid at the same time that we measure the
thermalconductivity. The motivation to measure the ther-mal
diffusivity is, of course, to obtain values of thespecific heat,
Cp. A description of the changes inthe system and initial results
on argon are given in[4,5]. Most of the changes made to the
apparatusimproved the measurement of resistance. The the-ory of the
measurement of thermal conductivity bythe transient hot-wire method
has been given in [6].For the measurement of thermal diffusivity,
thecorrections required by the theory had to be evalu-ated anew
[7]. It turned out, not unexpectedly, thataccurate measurement of
the wire resistance was ofthe utmost importance.
All of the changes and improvements were alsoincorporated into
our second apparatus, which wasdesigned to operate at higher
temperatures [3]. TheWheatstone bridge circuit, shown in figure 1,
waschanged to improve the accuracy with which thehot-wire
resistances and the initial balance condi-tion could be measured.
This was accomplished byadding a digital voltmeter to the system
capable ofmeasuring voltages to 0.5 ,V at the 200 mV level.The
voltages required in the wire calibration andbridge-balancing cycle
are fed to the voltmeterthrough a new multiplexer. Each arm of the
new
PowerSwitch
Figure 1. A schematic circuit diagram of the Wheatstone Bridge.
Potential taps areindicated by the points A-L.
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
bridge is about 200 ft at ambient temperature andincludes a
series of precise decade resistances. Be-cause the arms have higher
resistances than in theold system, it is possible to include a
calibrated 100ft standard resistor in each side of the bridge;
thusthe current in each side of the bridge can be mea-sured
independently.
The new high-temperature apparatus differs inseveral other
aspects from the low-temperatureone. Important to the present
discussion is the factthat in the new system there are seven leads
intothe cell rather than three. With the new arrange-ment of the
leads, shown in figure 2, it is now possi-ble to measure the
voltages across both the longand the short hot wires directly. This
eliminates theneed to account for (nuisance) lead resistances
andtheir dependence on temperature within the cell.The resistance
measurements are made as follows.With a supply voltage between 50
and 100 mV thecurrent in the left side of the bridge (fig. 1) is
deter-mined by measuring the voltage drop across thecalibrated 100
ft standard resistor at potential taps Iand J. The voltage drops
across the hot wires aremeasured between voltage taps E and F, and
G andH. The taps are shown in figure 1 while the physi-cal
arrangement is shown in figure 2. In summary,we can now measure
each resistance with an un-certainty of 9 mft which is considerably
better thanthe uncertainty of the earliest version of the
low-temperature instrument. The measurements de-scribed here for
nitrogen at 300 K with pressuresup to 70 MPa are the first to be
made with the newhigh-temperature system.
168.95-
0
0
168.85-
o 0o3) Long Hot-Wire
168.75
Bridge Point C-__1
.Bridge Point F-
Bridge Point GHLong IHot Wire
Short,H tWire
' Bridge Point H
Bridge Point E
-I High PressureP P P Cell Boundary
i i
Figure 2. Arrangement of current leads (i) and potential taps
(P)within the high pressure cell. Bridge points correspond to
thosein figure 1.
4. Measurements
Figure 3 shows the measured resistances of boththe long and
short hot wires as a function of thefluid pressure up to 70 MPa at
300 K. The resis-tance of each wire clearly decreases as the
pressureincreases. We represent the data with a straight linefor
each wire,
40 50 60
D.
43
Figure 3. Wire resistance for long and short hot wires as a
function of pressure for a temper-ature of 300 K.
115
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Journal of Research of the National Institute of Standards and
Technology
R(T,P)=R(T,O)+DP, or
R (T,P)/R (T,0)= I + y P. (2)
The resistance lines are shown in figure 3; thecoefficients and
standard deviations are:
R(300,0) D St. dev. (1 o-) yil f MPa-' n MPa-'
long hot wire 168.9085 -0.003 294 46 0.009 -1.95 X l0-5short hot
wire 43.0667 -0.000 870 63 0.003 -2.02x l0-5
In order to compare our results with those ofBridgman [1], we
compare the ratios R(300,70)/R(300,0). We obtained a ratio of
0.99864 for thelong hot wire and 0.99859 for the short hot wire.The
value interpolated from Bridgman's paper [1]is 0.9986. The
agreement with Bridgman's value isexcellent, and we conclude that
the measured resis-tance changes are caused by changes in the
fluidpressure. These resistance changes might also betemperature
dependent. From the wire calibrationestablished during recent
thermal conductivitymeasurements on nitrogen [8], which were,
how-ever, made in the low-temperature system, we ob-tained values
of 'y of - 1.85 X l0-' for 250 K and-2.2X 10-' for 100 K, a change
of about 20 per-cent in y.
[2] Roder, H. M., A transient hot wire thermal
conductivityapparatus for fluids, J. Res. Natl. Bur. Stand. (U.S.)
86, 457(1981).
[3] Roder, H. M., A high temperature thermal conductivity
ap-paratus for fluids, Proc. of the Fifth Symposium on
EnergyEngineering Sciences, Instrumentation, Diagnostics,
andMaterial Behavior, June 17-19, Argonne National Lab. Ar-gonne,
IL, Dept. of Energy CONF-8706187.
[4] Roder, H. M., and Nieto de Castro, C. A., Heat capacity,Cp,
of fluids from transient hot wire measurements, Cryo-genics 27, 312
(1987).
[5] Roder, H. M. and Nieto de Castro, C. A., Measurement
ofthermal conductivity and thermal diffusivity of fluids over awide
range of densities. Paper presented at the 20th Int.Thermal Cond.
Conf.; 1987 Oct 10-21, Blacksburg, Vir-ginia.
[6] Healy, J. J., de Groot, J. J., and Kestin, J., The theory of
thetransient hot-wire method for measuring thermal conductiv-ity,
Physica 82C, 392 (1976).
[7] Nieto de Castro, C. A., Taxis, B., Roder, H. M., andWakeham,
W. A., Thermal diffusivity measurements by thetransient hot-wire
technique: a reappraisal, Int. J. Thermo-phys. 9, 293 (1988).
[8] Roder, H. M., Perkins, R. A., and Nieto de Castro, C.
A.,Experimental thermal conductivity, thermal diffusivity
andspecific heat values of argon and nitrogen, Natl. Inst.
Stand.Tech., NISTIR 89-3902; 1988 Oct. 50 p.
5. Summary
We close with the recommendation that, if theultimate in
accuracy is to be obtained in a thermalconductivity measurement, an
in situ calibration ofthe hot wires should be performed. If an in
situcalibration of the hot wires cannot be performed,then the
resistance change of the wires can betaken as an additive,
calculated correction using ay of -2X l0-' MPa-', as shown in eq
(2).
About the authors: H. M. Roder is a physicist andR. A. Perkins
is a chemical engineer. Both are in-volved in the measurement of
thermal transport prop-erties of fluids in the Thermophysics
Division of theNIST National Measurement Laboratory at
Boulder,Co.
6. References
[1] Bridgman, P. W., The resistance of 72 elements, alloys
andcompounds to 100 000 kg/cm 2 , Proc. Amer. Acad. Arts Sci.81,
165 (1952).
116
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Scattering Parameters RepresentingImperfections in Precision
Coaxial
Air Lines
Volume 94 Number 2 March-April 1989
Donald R. Holt Scattering parameter expressions are de- the
conductor radii measurement preci-veloped for the principal mode of
a sion.
National Institute of Standards coaxial air line. The model
allows forand Technology, skin-effect loss and dimensional varia-
Key words: Bergman kernel; coaxial airBoulder, CO 80303 tions in
the inner and outer conductors. line; conformal mapping; coupling;
cu-
Small deviations from conductor circu- bic splines; error
sources; measurementlar cross sections are conformally contour;
measurement precision; princi-mapped by the Bergman kernel tech-
pal mode; scattering parameters; skin ef-nique. Numerical results
are illustrated fect; surface roughness; telegraphistfor a 7 mm air
line. An error analysis equations.reveals that the accuracy of the
scatter-ing parameters is limited primarily by Accepted: August 1,
1988
1. Introduction
To accurately characterize imperfections of pre-cision coaxial
air lines, skin effect and surfaceroughness need to be considered.
Skin effect is nowwell documented [1] and conductor surface
finishhas been studied in detail by Rice [2] and Ament [3]through
the use of Fourier series methods. WhileKarbowiak [4] points out
that Fourier analysis re-veals useful knowledge of the spectral
componentswhich principally affect scattering parameters, it isalso
appropriate to examine local pointwise influ-ence along the axial
(z) coordinate. In this connec-tion, Hill [5] developed
perturbation expressionsfor the scattering parameters-for a
lossless circularair line. When the conductor surface exhibits
trans-verse angular variation, Rouneliotes, Houssain andFikioris
report the effects of ellipticity and eccen-tricity on cutoff wave
numbers [6].
The purpose of this paper is to develop numeri-cally accurate
pointwise coaxial air-line scatteringparameters that account for
skin effect loss andconductor surface variations in the transverse
an-
gular and axial directions. Following Schelkunoff[7], Reiter
[81, Solymar [9] and Gallawa [10], gener-alized telegraphist
equations for the principal modeare derived in section 2 for a
circular air line.Transformation to forward and backward
wavedifferential equations enables general solutions forthe
scattering parameters in section 3. To allow forconductor surface
measurements along the z-axis,cubic spline polynomials provide a
starting pointfor establishing pointwise recursion formulas
offorward and backward waves in section 4. In sec-tion 5, the
Bergman's kernel technique is used toestablish a conformal mapping
for transformingnoncircular conductors into equivalent
circularconductors in correspondence to the principalmode.
Computational results illustrating I Si, I ver-sus air-line length
are given in section 6. An erroranalysis of the computational
algorithms for the ac-curacy resolution of the measurement system
is de-veloped in section 7.
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2. Generalized Telegraphist Equations forthe Principal Mode
Consider the coaxial air line in figure 1. Withinner radius a(z)
and outer radius b(z) the fieldcomponents of primary interest are
radial electricfield (E,), angular magnetic field (Ho), and
axialelectric field (Ez). We assume the fields E,, Ho, andE, are
composed of TEM and TM modes, and cou-pling of the modes is caused
by skin effect withvariations of the conductor surfaces. From
Ap-pendix A boundary conditions for TM modes pos-sess the form
E =-ZsK{4b(z)}Ho atr=b(z),
Ez= +ZsK{4fa(Z)}Ho at r=a(z),
where for instance,
tions a set of orthogonal basis functions needs to beconstructed
from the Gram-Schmidt process. As-suming E,, Ho, and E, possess
continuous first andsecond derivatives implies their expansions are
ab-solutely and uniformly convergent [12]. In Ap-pendix B these
properties are used to rearrange theexpansions into the form
(aorw)
E,(rOz)= 2 (z) e(( (rOz)(n,p)= (0,o)
+ Vp) (z) e (2) (r,0,z) (1.5)
(.1.)
H(rOz)= : 1('p)(z) h~n(p (rOnz)(n,p) = (,O)
+1(2p)(z) hpn,^2 (r,O,z)
(1.0)
(1.1) (1.6)
where the superscripts (1) and (2) represent evenand odd modes,
respectively. We have
K{+b(Z)} a+sin [b(Z)]W (1.2)
Appropriate Maxwell equations for determiningtransverse fields
Er and Ho in the air dielectric re-gion of the air line are
[11]
aE_=____ to+ aEz (1.3)az ar
aH0o - -jctJe Er. (1.4)az
The parameters &), p., and e are defined as radianfrequency,
permeability and permittivity, respec-tively, In addition the
fields are assumed to varywith time according to the complex
exponentialfunction eP@t.
To find the generalized telegraphist equations itis convenient
to assume the fields possess orthogo-nal expansions in r and 0. In
view of TEM and TMmodes together with impedance boundary condi-
{ern (r,O,z)1 1 cos kOe(2) (r,6,z)J =N-p(z)fnP(rrz)l sin k }.-
(1.7)
In addition,
ao heS (r,Oz)= az Xar e(?) (r,lz); i= 1,2 (1.8)
and Nnp(z) denotes the norm of frnp(rz), that is,
1-b(z)
Nnp(z) =I {fnp(rz) frnp (rsz)}l"2dr,a(z)
(1.9)
where - stands for the complex conjugate. In par-ticular for the
TEM mode
ePOo(rz)=No(z) r ' (1. 10)
No(z) = 12,x In b (z) } 112.
Higher order modes are usual linear combina-tions of the first
derivative Bessel functions Jn andY',-
Following Reiter [8] by taking the inner productof eq (1.3) with
the basis function ek) yields
T 'Er 6 e&dS = -jct)p f aHo - {aohWt4 X az dSS(z) S(Z)
+ aEze5,dSS(z)
(1.1 1)
Figure 1. Coaxial air line. where S(z) denotes the cross
sectional air dielec-tric region between the conductors. The left
side
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
calls for differentiation of a variable surface inte-gral and
the second member of the right side inte-grates by parts.' Hence,
eq (1.11) evolves into theform
dz E r qdSjf E, I Ei)4dS5(Z) .5(z)
= Ltan{Ob (z)}E, r k) ds
-f Ltan{4a(z)}E, ePqds H0 h-f o dS5(z)
+ L{Ez[b (z),Oz] -Ez [a (z),O,z]}?,k)qds
- r E , qdS. (1.12)5(z)
To express E_ in terms of the constructed basisfunction (fnp)
let the component of E. correspond-ing to k =0 (TMo, modes) be
expressed as
Eo=7r - bop(z) fop (u) du.p=l -J
(1.13)
From Maxwell's equations the (Op) mode relationbetween Er and Ez
is
roP~rp)=-k2p ar
Ig(Z)~Jfo$a(z),z] 27r - k'0o Vg,(z)p=O yow(z) Noo(zp= 1 yop No0
(z)
Cb(z)CaJ r)fop J) du - foo(rz) r drNoo(z) ()J Np (z)
-27r tan[4a(z)] .oo [a(),z] fzz (z) ,z] VJ z (z).-27r tan[a
Noo(z) a(z)j N o(z) (1.16)
To derive the companion generalized telegraphistequation from eq
(1.4), the procedure is almostidentical. Taking the inner product
of eq (1.4) withhIk4) from eq (1.8) yields
,aH °A, dS =-IcEr aoEr -az X aerk), dS.S(z) S(z) (1.17)
For the left side
f ao hW4 dS = dIkq kp fSd aX0 S(z)
d ~az X a~rkl) dS ta ~1b(-~ OgR,(1.14)
+ , tan {4a(Z)} Ho Qlq dS. (1.18)where Yop and kcop are the
propagation constant andcutoff frequency numbers, respectively.
Using eqs(1.5), (1.13), and (1.14) yields
(1.15)bop (z) = * Nn (z) Vgp)(z).
Now substituting basis function definitions eq(1.7) and calling
for the principal mode yields
dVoo_ fo2 (rrz) rA1)Iaz NooIZ)ddz azNO( ) FoPZazNo(rz)
= jco~u oZS K[4 b(z){ (z),z] b(z)=-jwp.192(z)-27TZ N[b 0)Jj(z
bz)
fo, b (z),z]P=O $P)(z) Nop(z)
-2irZsK[4,a(Z)] a(z), a(z).
1 Since S(z) is differentiable and the field components are
con-tinuous, interchange of differentiation and integration is
justi-fied.
Since integration by parts obtains the relation
T ,) d C 1) d e g ) dS,r e Pp dd e(rq dS = j , e(rp dS,S(z)
S(z)
(1.19)
eq (1.17) takes on the form (setting k=q=0)
dIo o
dz p=o5(z)I
651, dd e(',) dS
+b'(z) OL Ho Q dS -a'(z) fiL Ho ho dS.(1.20)
Substituting eqs (1.6) and (1.10) into eq (1.20) gives
+ 2 I$(z) 1 °°z a f(rZ) r dr=OpkZ ooA(z,\ az N0 p(z)
= -jcie VgO(z)
+ 21r b'(z) A b (z) S $(z) fojb(z),zNoo(z) Nop (z)
-27T a'(z)°[()'] a (z) I(z) fo, a (z)) Noo(z) p=O Nop(z)
P=O ~~(1.21)
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Examining eqs (1.16) and (1.21) reveals that con-tinuous mode
coupling occurs through the voltageand current transfer
coefficients (left side), respec-tively, a phenomenon observed by
Schelkunoff [7].Skin effect coupling on the conductor surfaces
wasalso reported by Schelkunoff and Gallawa [10].When the air line
is operated at frequencies appro-priate to the principal mode, all
TM modes attenu-ate rapidly below their cutoff
frequencies.Consequently, dominant coupling occurs betweenthe
forward and backward waves of the principalmode.2 In this regard,
eq (1.16) assumes the form
d Voo b(z) I lId{f_2X JJ .\.1/ 2 -adz 27arz Inlnukz) r az
LS[ ~a (z)J
az)]1/2L2 n b) drj VOO(z)=-job 1OO(z)
-Zs{Kf[I b(z)] 1b(.) l
a (z)
+KkfOa(Z)] 00Za (z) In ab(z)l
+ b'(z)b (z) In bz
a (z)
ab (z) VOO(Z),b (z) Inbzl
a (Z)
(1.22)
where the superscript, (1), has been dropped sinceonly one mode
is involved. Rearranging terms pro-duces the expression
d Voo +QHJi+Z 3 K(z)}Iao==Too(')(Z) V0O, (1.23)
where the current transfer coefficient is defined as
(6 z 31 0b'(z) _2b (z)lInb (z)
a (z)
a'(z)}a (z) In a(Z)
(1.27)
3. Conversion of Generalized TelegraphistEquations to Forward
and BackwardWave Equations
Following Solymar [9] we define the amplitudesof the forward and
backward waves A 0+0 and A 0-,from the relations
VOO=kOU/2{A d++A -},
1oo=ko- "2 {A 0+-A -0},
where ko=tl/E, the wave impedance. Substitutingeq (2.1) into eqs
(1.4) and (1.20) produces the ex-pression
dA o + ZK(Z)1A + =TzA+dz + 2ko IA 00 ooz 00
ZK(Z) A o- Too(z) A (2.2)2k0 02
dA - [.ZSK(Z)1 _00 VA+2k 1A o-= Too(z) A -
_K___ A o- Too(z) A + (2.3)2kw 2 =0.uwhere I6= cAt)Le,
Too(z) = {T~2(z) + TT(z)}.where
(2.4)
_K[4b(z)] I 1b(z) + a(z) 1 b(z)'
a (z)
TM()- = I b'(z) _a'(z)l 12{ b(z) a(z) Jln b(z)
a (z)
In view of eqs (1.25) and (1.27) the last expression(1.24)
possesses the form
(1.25)
The equation for current proceeds similarly. Equa-tion (1.20)
yields
dz- = -]cE Voo+ T((z) loo (1.26)
2 Higher order mode influence on the TEM mode will be re-ported
in a later issue of this journal.
T00(=ln b(z) l b (z) a (z) Jia (z)
For a lossless airline, voltage and current trans-fer
coefficients assume the form,
- T2(z = ~(z =1 1z Ib'(z) __ a'(z)l-T~~o~) =T~o°(Z =2 In { b (z)
a (z) }'
a (z)(2.6)
120
(2.5)
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Also, using Solymar's assumption that reflection ofthe principal
mode does not affect forward propa-gation of the principal mode
yields expressions
dA °° +ijB A =o0, (2.7a)dz 0
and
dA+. A 1 1 fb'(z) a'(z)jdz JiJ =21 b(Z) bn (z ig
a (z)(2.7b)
which agree with Hill's results [5].Returning to eqs (2.2) and
(2.3) and retaining
Solymar's assumption above leaves the terms Too(z)A '0+. Since
coupling in this sense is meaningless, wedrop the terms Too(z) A
O0+ and obtain 3
dA oo + 0+(Z.K(Z) A + =000 2 k j; , - s W 'I 00 (2.8)
4. Cubic Spline Fitting of ConductorRadius Measurements
Underlying an accurate solution to A &+ and A -are two
critical items: (a) fitting conductor radiimeasurements with
acceptable error bounds and(b) expansion of all known functions in
a systematicmanner to sufficient powers of z.
To handle (a) consider cubic spline polynomials[13] for the
inner (or outer) conductor measure-ments such that
Ck- 1(Z) = 1ok- 1+- *+ C3,k-l Z' (3.1)
where Ck_1(z) approximates a(z) or b(z),Zk- iZ
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
2ko + 2 =i n Sk,k=0where O•
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Volume 94, Number 2, March-April 1989
Journal of Research of the National Institute of Standards and
Technology
Then the length of L is
°(r; dS=f; {p2(O)+[@]21 dO (4.5)To find representations for the
Szeg6 polynomials,
consider the following orthogonalization proce-dure. Let the
matrix elements,
hpq= IL XqdS= I { pP+q(O){p2 (O)
+ [p]21 ei(p-q)O dO, (4.6)
be defined for p >0 and q >0. Note Apq =h,. Thenfollowing
Kantorivich and Krylov [15] computethe determinant
hooDo=1,Dn=
hon
(4.7)hio ... hno
hIn ... hnn
Hence, the Szego polynomial is defined as
(4.8)
1 hooPn(0) [D._IDj 1,12 hol.
ho, n-l
such that
A convenient property of the Szeg6 coefficientsfor symmetric
contours is found from eq (4.6). Wehavehpq = f r {l +(- l)p+q(O){p
2(O)
e 0
+ [ap] 1 1/2e(P-q)Od.
For off diagonal elements
hpq=O;p+q odd,p-7q.
Hence, an