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NF-791-1
SMOOTHED THERMOCOUPLE TABLES
E.G. Davidson and R.K. Adams
Oak Ridge National Laboratory
Oak Ridge, Tennessee
MASTER
ABSTRACT
, -
-
:::
Since the rounding error associated with the present standard
tables for thermocouple emf versus temperature is too large to allow
the use of these tables for precision thermometry, the authors have
obtained smooth tables which are expressed to a significance of
0.001 microvolt and which witch the NBS Circular 561 tables every-
where to vith in the NBS sounding.
The method and mathematical techniques for preparing the tables
and the results are described. The address 18 given where punched
card copies of the smoothed tables or fortuna computer programs
fot o s may be obtained.
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INTRODUCTION
.
1
:
.PARA
For several years tables of temperature versus emf published in the
National Bureau of Standards (NBS) Circular 561" for standard thermocouple
-
F
materials have been, and continue to be, widely used as the recognized
standard on which many precise thermocouples are based. In these tables,
S
tenperature versus enf 18 tabulated at every degree within the range of the
Research sponsored by the U.S. Atomic Bergy Coumission under contract
with the Uaion Carbide Corporation.
PASIN Standards, 1964, Part 30, includes thermocouple tables based on
values in the NBS Circular 561. This paper references NBS Circular 561,
although the tables in the ASIN Standard are essentially identical.
.
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thermocouple material to an accuracy of 1 microvolt in some cases and 10
microvolts in others.3 Although the error associated with this degree
of accuracy 18 acceptable for most industrial applications," and the accuracy
of temperatures determined from the emf of an uncalibrated thermocouple
rounding
does not require smoother tables, the error 18 too great for certain
laboratory experiments, particularly for those involving small or measure-
ments where precision thermometry 18 required. A portion of Table 2 from
. Further, it
NBS Circular 561, plotted in Fig. 1, 1llustrates this point. ve plot
the deviations of the table entries from a smooth curve, which might
represent a real thermocouple, we obtain a curve as shown in Fig. 2. For
, -.
-
:::
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laboratory experiments in which the slope or the change in slope of temp-
-
erature with respect to some other physical variable is the important
measurement, it is desirable for interpolation purposes to have thermocouple
tables of extended significance.
Such tables should be smooth to a degree
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of significance such that the discontinuities due to rounding error do not
obliterate the small changes that the experiment was designed to exhibit.
Digital computer processing of experimental data involving temperature
requires brief mention of another consideration. A table of emf versus
temperature requires significant computer memory for storage, whereas a
formula does not. A formula 18 easier to program, but table lookup 18
usually faster.
.
H. Shenker et al., Reference Tables for Thermocouples, NBS Circular
**, .s. Dept. of Commerce (April 27, 1955)
ISA standard tolerances and ASIN. limits of error are adequate for
most industrial work.

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TECHNIQUES FOR SMOOTH ING THERMOCOUPLE 'TABLES
The problem of smoothing the NBS tables has been approached by others
using different techniques. One technique involved plotting first dirferences
with visual cut-and-try smoothing,'followed by integration of the smoother
difference curve and successive adjustment and re-integration to minimize
the difference between the integrated emf's and the NBS table entries. An
attempt was also made to keep the second differences from being discontinuous.
This technique buffers from being slow and subject to human errors.
Another technique used Lagrange second degree polynomials for interpolation
between key temperatures. This technique, although it provides a result
that requires! 1688 computer memory, suffers from being discontinuous in its
first and second derivatives at the junction points between polynomlale.
*
!
GE
There are undoubtedly other techniques which have been used, most of
which either follow the previous illustrations or are similar to some we
tried and found inadequate.
In our first attempt to smooth the tables we tried to fit a polynomial
to the tables by the method of least squares, using a 10th-degree polynomial
for the platinum-platinum-10% rhodium tablep with data points at every sec.
The difference between the computed and tabular values was plotted for every
degree within the range of the table (Fig. 3). A similar plot was made
(F48. 4) for the same table, using a fourth-degree polynomial with an
added exponential teru. This formula vas obtained from Mr. R.D. Fynn of
53.7. Potts, Jr., and D.L. McElroy, Thermocouple Research to 1000°C -
Final Report Nov. 1, 1957, through June 30, 1959, ORNL-2773, p 214.
°R.P. Benedict and 8.F. Ashby, Empirical Determination of Thermocouple
Characteristics, ASMS-61-WA-19 (1961). Also, "Improved Reference Tables for
Thermocouples," p 51 in Temperature, Its Measurement and control in Science
and Tndustry, voi. 3, Part 2 (1962).
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tne Heat Transfer Section at NBS and is based on a theoretical thermocouple
model.
Its use was not recomended for temperatures above 14.50°C for reasons
that are obvious from the figure.
These curves show: (1) the sharp, l-microvolt-high, discontinuities
(sawteeth) are due to round ing error in the tables; and (2) the longer
undulations show the ability (or inab111ty) of the formulas to match the
tables.
Clearly, these formulas fail to match the tables by a wide margin
compared with the magnitude of the rounding error. About all that can be
.
1
said for the formulas 18 that they are continuous. To make the polynomial
function fit the tables and to have an error of about the same order of
magnitude as the rounding error would seem to require a polynomial of a
rather large degree. For this reason, a smoothed table of emf versus
temperature 18, in general, the only adequate representation of thermocouple
characteristics for scientific work, and the digital computer memory
required to store the tables for automatic calculations simply must be
tolerated.
DIFFERENCE CURVES VERSUS SMOOTHED TABLES
Calibration curves for
Mercuremontidades de specific thermocouple, usually do not follow the
- Characteristic curves you
NBS tables exactly, of course, and the date on different batches of wire
deviate from the tables in different manners. Although a polynomial
representation of the tables did not appear to be feasible, the same cannot
be said of a polynomial representation of the difference curve' associated
with a specific thermocouple. The argument 18 simply one of magnitude.
'W.F. Roeber and S.T. Lonberger, Methods of Testing Thermocouples and
Termocouple Materials, NBS Circular 590, V.8. Dept. of Commerce (Mb. 6, 1958).
.
a5
The difference curve for a specific thermocouple 18 psually expressed in
microvolts, but the tables are expressed in millivolts. Although the relative
error in representing either the tables or the difference curve by a poly-
nomial would be the same in either case, the absolute error 18 a factor of
1000 smaller for the difference curve. The number of calibration points
for a specific thermocouple 18, on the average, about 81x, which 18 sufficient
to determine a low order polynomial representing the difference curve. To
determine a polynomial of the nth degree in the least squares sense requires
a minimum of (a + 1) pairs of points. The difference polynomial 1s con-
tinuous in all derivatives and may be computed to any degree of significance.
Howeverbefore the difference polynomial can be derived and before the
polynomial can be used in thermometry, we need a basic table with the
necessary significance that follows the major excursions characteristic of
the material of the thermocouple. Thus, an NBS table Amoothed to a factor
of 100 or 1000 greater significance 18 needed.
MATHEMATICAL TECHNIQUE FOR SMOOTH ING
To explain the technique used in smoothing the tables consider the
following set of data drawn somewhere from within the table. For purposes
ميم مهنا
of discussion we will arbitrarily let it consist of five voltages associated
with five adjacent temperatures (refer to Fig. 5). To compute a better
(smoother) value of the voltage for the center temperature in the group of
five, we take all five voltages and use them to determine in the least
squares sense a low ordur polynomial of, for instance, the second degree.
This polynomial y de evaluated at the center temperature and, thus, we
determine a better (smoother) value of the voltage to be associated with
-
---..
.
..........
:
6-
the center temperature. It will be close to the original value of the voltage
at the center temperature, but not necessarily the same value. Liketise, we
may determine a better value for every voltage in the table by successiveiy
fitting a polynomial and calculating the center value. This 18 done by
moving along the table, dropping the first point and including the next
point in the table.
The lowermost two entries and the upupermost two entries
will require special consideration, since they do not have both two bigher
.
buy wie se
neighbors and two lower neighbors. In the case of the two lowermost voltages,
& smoothea value for them 18 computed, wing the same polynomial that was
determined for the third point from the end. It 18 simply evaluated at the
-
-
-
-
-
-
-'.
.
appropriate lowermost two temperatures for which a smoother value of the
ALE
voltage is desired. Similar considerations apply to the uppermost two
voltages. Once these smoother values have been computed, the whole operation
may be repeated by operating on the previously computed voltages and returning
an even smoother set of voltages. Those familiar with statistical techniques
will recognize that, if the approximating polynomial in the previous discussion
18 of zero degree (in other words, the function is approximated by & constant
which is nothing but the average of the five points), we are employing the
technique long used by statisticians for smoothing statistical time series.
They refer to it as the "moving average" technique.
Before applying this procedure, we must answer three questions: '
1. What degree polynonial should be used.
2. How many times 18 it necessary to repeat the smoothing operation to
achieve a given degree of smoothe88.
3. How many data points should be included in the group for smoothing.
The first question need de answered the only on approximate agere before
ve ans apply the technique. We first construct a table of differences of
.
To
several orders from the original data. To do this, we find it convenient
to first normalize the data, that is, we multiply the data by an Appropriate
.
power of 10 and convert the result to integer form, such that the least
:
...
Bignificant digit in the integer form corresponds to the last digit of
...
significance in the decimal form. From these normalized data, differences
.
.
out to several orders are computed and examined.
.
.
- ....
To interpret these data, let us first review some properties of
difference tables. If the original data had been derived from a polynomial
of the ath degree, then the (a + 1)th order and all higher orders of
differences would be zero. If a difference table 18 constructed from a
function that 18 not a polynomial, it will be found (1f the function is
tabulated at sufficiently close intervals) that the second-order differences
are smaller in absolute value than the first-order differences, that the
third-order differences are smaller than the second-order differences, and
so on, until an order is reached at which the differences are small whole
numbers and alternate randomly in sign. Beyond this order, the differences
aga in grow in absolute value but continue to alternate in sign. That order
of differences whose magnitude was smallest may be likened to the (a + 1)th-
ordered differences for the polynomial. In other words, if a polynomial 18
chosen to represent the function over a short interval, a polynomial of
the ath degree cou] 1 best represent the function.
This provides an approximate
answer to the first question.
The answer to the second question also involves a difference table.
Suppose we have selected the degree of the polynomial and the nubber of data
points to be included in the group for smoothing, and we have actually smoothed
the data once. Let us further suppose that we want our table to be smoothed
out to the closest 0.01 microvolt. If the smoothed data are now truncated
at the 0.01-microvo.lt level and are normalized, and a table of differences is
-
constructed from this data, we will again find a given order of differences
whose absolute values will be less on the average than for any other order,
and these differences will alternate randomly in sign.
If the differences
-.
.
.-
.
are, for the most part, 0 and £ 1, with maybe an occasional + 2, we may
conclude that the previous smoothing operation was sufficient. If the above
condition does not hold, we must repeat the smoothing operation until it does.
Let us now turn to a consideration of the third question. The answer
.
-
to this question is related to the first question and for this reason, the
previous discussion gave only an approximate anqwer. The number of data points
selected for smoothing could be as few as three. An odd number is desirable for
symmetry reasons. The ability or a polynomial of given degree to approximate
the original data will depend on the size of the interval and, consequently,
the number of data points used for smoothing.
The shorter the interval, the
more faithfully the smoothed data will follow the original data.
This is
not necessarily desirable, because the original data presents a saw-toothed
pattern relative to the true function that it supposedly represents. To
overcome this difficulty and to prevent tbe smoothed data from retaining the
saw-toothed nature of the original rounding beer pattern, it is desirable to
include sufficient data points in the smoothing group to bridge the longest
span between adjacent teeth in the saw-tbbthed pattern. This saw-toothed
pattern 18 best seen in a plot like Figs. 3 and 4, but it may also be obtained
from examining the difference table of the original data. The span between
AR
***
teeth 18 as large as 90 degrees in one case. Over intervals of this size,
1t 18 necessary to reexamine the answer to the first question. It may be
desirable to raise the degree of the approximating polynomial.
After this technique has been carried out, it will be found that the
smoothed data curve does not pass exactly through the origin, as it must
if it is to represent a real thermocouple. This little wrinkle 18 overcome
by subtracting the residue at the origin from all the entries in the smoothed
table.
If the smoothing has been gone with close attention to all three
iti
questions discussed, the magnitude of the residue at the origin will be
#
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*
less than the original rounding error.
Increme sp
*
DISCUSSION OF RESULTS
Table 2 from
thead
The result of smoothing NBS Circular 561 tabtes using this technique is
***
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**
.
shown in Fig. 6.
A portion of the smoothed difference table is shown in
Table 1.
-- ----. tom
Recently, NBS has employed digital computer techniques in its type *
--
...
thermocouple calibration work, but has continued using the segmental poly-
8
nomial technique first published by Roeser and Wenzel.
The coefficients
of their original eight therd-degree polynomials were designed for the
International Temperature Scale of 1927 and for emf's expressed in Inter-
national volts. Based on these polynomials, with adjustments for the
International Temperature Scale of 1948 and emf's expressed in absolute volts,
we calculated a smooth table, designated NBS 127. A plot of the difference 6
between the smoothed type #table obtained from NBS Circular 561 and NBS 127 18
shown in Fig. 7.
*W.F. Roeser and H.T. Wenzei, Reference Tables for Platinum to Platinum-
Rhodium Thermocouples, NBS Research Paper No. 530, U. 8. Dept. of Commerce (Feb. 1933).
-10-
At present, NBS 18 using a seven-polynomial "table" in their computer
calculations; the coefficients of the seven polynomials currently beiug used
are
by NBS s given in Table 2 of this paper. Figure 8 is a plot of the dif-
ference between the smoothed Table 2 of NBS Circular 561, as determined by
the authors, and a curve determined from these polynomials (designated
NBS T48).
The primary advantage in using a smoothed table instead of a set of
polyromials in precision work is the speed of conversion from emf to temperature
in the computer. The smoothed tables from NBS Circular 561 have an additional
advantage: they are based on long-used and well-established tables which,
whether or not they well represent actual thermocouple characteristics, are
used by nearly all who work with thermocouple materials in the United States.
Also, as is desirable for the several reasons mentioned earlier, the
derivatives (through the second order) of the smoothed table are nearly
continuous functions, whereas only the first derivatives of the NBS set of
polynomials are designed to match at the junction points between polynomials.
The authors' procedure has been applied to several types of thermocouple
materials (Table 3).
The approximating polynomial was of the second degree
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Punched card copies of the smoothed tables pour
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can be obtained by writing to Mr. F.C. Vonderlage,
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Office of Industrial Cooperation, Oak Ridge National laboratory, P.O. Box X,
Oak Ridge, Tennessee. An Oak Ridge National Laboratory report, ORNL-3649,
Vol. I, documents the entire investigation of thermocouple tables, and Vol. II
contains the smoothed tables in printed form. These two volumes can be obtained
from the Division of Technical Information Extension, P.0. Box E, Oak Ridge, Tennessee.
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Smoothed Difference Table
Processing of Nes Circular 56),
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DIFFERENCE ORDER
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Table 2.)
Rocfficients* of Polynomials for "ype 8 Thermocouple
e.m.f. = A + BT + Cm ? + DT2
where ent is Absolute Microvolts, T in deg. C.
International Practical Temperature Scale of 1948
Temp. Range
et
0 - 231.90
231.90 • 327.40
327.40 - 419.45
419.45 - 630.50
630.5 • 1063.0
1063 · 1250
5.4291221 1.1045780 x 10-2 - 1.1624775 x 105
6.0813770 7.3742742 x 20-3 - 5.1125365 x 100
6.2766721 6.7056532 x 10-3 - 4.3585867 x 10-6
8.0100051 2.0856326 x 10-3 - 2.9964276 x 10-7
8.1229839 . 1. 7126453 x 103
3. 7680664 6.3245099 x 103 -1.6014100 x 100
- 53.767999
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Based on private communication from abs, 5/21/64.
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PABLO III
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Cosmoothei Thermocouple Tables Based on NBS Circular 561. All tables are expressed: to
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-LEGAL NOTICE -
This report was prepared as an account of Government sponsored work. Neithor the United
States, nor the Commission, nor any person acting on behalf of the Commission:
A. Makes uny warranty or ropresentation, expressod or implied, with respect to the accu-
racy, complotoness, or usofulness of the information containod in this roport, or that the uso
of any information, apparatus, molhod, or procoso disclouod in this roport may not Infringo
privatoly owned righto; or
B. Assumos any liabilities with rospect to the uso of, or for damages resulting from the
use of any information, apparatus, molbod, or process disclosed in this report.
As usod in the abovo, “porson acting on behalf of the Commission" Includes any om.
ployoo or contractor of the Commission, or employco of such contractor, to the extent that
such omployoo or contractor of the Commission, or omployee of such contractor prepares,
disseminates, or provides access to, any Information pursuant to his omployment or contract
with the Commission, or his emplcyment with such contractor.

.
MY
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END