NBS 125

c71d0e78-ccf0-42dc-b7a9-50c49af5444c 11 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Part: 5

]]>

T/C types T, J . . . NBS Monograph 125

Both fits result from reduced.

]]> NBS 125

T 6.25629 - 268.935 - 6.22919 - 259.340 - 6.19773 - 252.870 - 6.17138 - 248.595 - 6.15358 - 246.048 - 5.87302 - 218.789 - 5.75328 - 210.002 - 5.53559 - 195.802 - 5.31472 - 182.962 - 2.74070 - 78.476 - 1.43494 - 38.862 - 00 1.0679 26.87 4.2773 100 5.3414 122.37 7.0364 156.634 11.0133 231.9681 13.2188 271.442 16.0953 321.108 16.4733 327.502 18.2179 356.66 212 mat :

J 8.0957 - 210.002 - 7.7963 - 195.802 - 7.4807 - 182.962 - 3.7187 - 78.476 - 1.9063 - 38.862 - 00 1.3739 26.87 5.2677 100 6.4886 122.37 8.3743 156.634 12.5517 231.9681 14.7427 271.442 17.4928 321.108 17.8462 327.502 19.456 356.66 22.9259 419.58 24.3123 444.674 30.1095 548.23 34.9108 630.74 36.6933 660.37 55.669 961.93 61.716 1064.43 62.880 1084.5 232 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

T T . 12 blacj plot : J J . 12 red plot : 21 sys

T J 2 1 sys

T/C type T . . . NBS Monograph 125

A most remarkable fit from data reduction

]]> Utilities

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - range V V stem j el stack : for V V data data rows row stack : 71 line :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

T 6.25629 - 268.935 - 6.22919 - 259.34 - 6.19773 - 252.87 - 6.17138 - 248.595 - 6.15358 - 246.048 - 5.87302 - 218.789 - 5.75328 - 210.002 - 5.53559 - 195.802 - 5.31472 - 182.962 - 2.7407 - 78.476 - 1.43494 - 38.862 - 1.0679 26.87 4.2773 100 5.3414 122.37 7.0364 156.634 11.0133 231.9681 13.2188 271.442 16.0953 321.108 16.4733 327.502 18.2179 356.66 202 mat :

X T 1 col : Y T 2 col : n T rows : 31 line

The data set is Type 'T' T/C [Thermocouple] from NBS Monograph 125. A damned good fit ! Former NBS is now NIST. Check Pearson stat. NI [National Instruments] were "delighted" from the method to personalise T/C [Leslie]

Treduced X Y X / vectorize augment :

jmG Mathcad DAEP [2003] β 25.8149913045858 12.7668104944864 - 1.25727409574702 3.04185012250532 0.611123673599964 0.040154623061311 0.00064384557168733 0.46562548019054 - 0.030977484107051 0.119835340769071 0.027040212533505 0.002100716189884 0.00004883292991218 131 mat :

Treduced o 4 black plot 'c' coefficients conserved 12 decimals

c 2.06616508306782 108 -^* 1.04086361374117 104^* - 337.097030645294 - 556.796817063866 23.5682134277668 1.24287589943351 - 1.66276414822246 0.18965762016879 3.34581525265509 103 -^* 402.356136811016 - 25.1336669438927 - 21.753666487766 1.52017639703213 0.050935169713453 - 0.0639853127034 9.23623646847396 103 -^* 2.57197875267178 104 -^* 171 mat :

c 2.0662 108 -^* 10408.63613741 - 337.097030645294 - 556.796817063866 23.568213427767 1.242875899434 - 1.662764148222 0.189657620169 0.003345815253 402.356136811016 - 25.133666943893 - 21.753666487766 1.520176397032 0.050935169713 - 0.063985312703 0.009236236468 0.000257197875 171 mat :

recover from 
reduced  Padé x T x c 1 el c 2 el x * + c 3 el x 2^* + c 4 el x 3^* + c 5 el x 4^* + c 6 el x 5^* + c 7 el x 6^* + c 8 el x 7^* + c 9 el x 8^* + 1 c 10 el x * + c 11 el x 2^* + c 12 el x 3^* + c 13 el x 4^* + c 14 el x 5^* + c 15 el x 6^* + c 16 el x 7^* + c 17 el x 8^* + / * :

Conformity from absolute residuals δ i 1 T rows range Δ i el Y i el X i el T - 3 round eval : for X Y Δ augment 21 line :

FIT U 6.26 - 18.25 6.2 - range : i 1 U rows range FIT i el U i el T eval : for U FIT augment 31 line :

δ 6.2563 - 268.935 - 0.001 6.2292 - 259.34 - 0 6.1977 - 252.87 - 0 6.1714 - 248.595 - 0 6.1536 - 246.048 - 0 5.873 - 218.789 - 0.001 5.7533 - 210.002 - 0.002 - 5.5356 - 195.802 - 0.001 5.3147 - 182.962 - 0.001 - 2.7407 - 78.476 - 0 1.4349 - 38.862 - 0 1.0679 26.87 0 4.2773 1000 5.3414 122.37 0.001 7.0364 156.634 0.002 - 11.0133 231.9681 0.005 13.2188 271.442 0.006 - 16.0953 321.108 0 16.4733 327.502 0.001 18.2179 356.66 0203 mat

data T . 12 black plot : datum X 0 Y * augment . 12 black plot : 21 sys

Rational fraction to Type 'T' T/C, NBS Monograph 125

data FIT 2 1 sys

Δelta X δ 3 col augment : stem Δelta Error : bar Δelta 0.075 BarSize : 31 line

0 stem datum 3 1 sys

Calibrate the 'T' mV/I converter

XmV 100 - :

XmV °C ≡

mV 1 :

mV T XmV - mV 5 - 2 - solve mV 3.3787 -

.

3.3787 - T 9.2866 T 21 mat 100 - 20021 mat

XmV 200 :

XmV °C ≡

mV 1 :

mV T XmV - mV 811 solve mV 9.2866

More statistical insight

SSD = "Sum Square Differences" SSD Y i el X i el T - (2^(i 1 n sum : 7.842632 105 -^*

χ X Mean : μ Y Mean : 21 line

χ μ n 31 mat 1.754059 14.306345 - 2031 mat

Pearson correlation

Pearson’s correlation coefficient corr X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el T eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el T eval μ - (2^(i 1 n sum * sqrt / / : 1.0000001

correlation between [X,Y data]

correlation between [X,fitted model]

X i el χ - (Y i el μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum Y i el μ - (2^(i 1 n sum * sqrt / X i el χ - (X i el T eval μ - (* (i 1 n sum X i el χ - (2^(i 1 n sum X i el T eval μ - (2^(i 1 n sum * sqrt / 21 mat 0.98633798 0.98633792 21 mat

T/C type J . . . NBS Monograph 125

A most remarkable fit from data reduction

]]> 'c' normalised coefficients DO NOT truncate

c 19.8518394369938 3.39708430077966 0.052276519294839 8.91649190043396 103 -^* - 9.92828185332406 105 -^* 8.38053206800752 106 -^* 3.1464248690302 107 -^* - 4.25243756863902 109 -^* 2.07174891774966 1011 -^* - 0.183112912617708 3.87083881009766 103 -^* 5.08713285834413 104 -^* - 2.26745307368593 106 -^* 6.6275111583499 107 -^* 2.21691344604783 108 -^* - 2.87497042511128 1010 -^* 1.36525976453514 1012 -^* - 171 mat :

recover 
reduced x J1084 x c 1 el c 2 el x * + c 3 el x 2^* + c 4 el x 3^* + c 5 el x 4^* + c 6 el x 5^* + c 7 el x 6^* + c 8 el x 7^* + c 9 el x 8^* + 1 c 10 el x * + c 11 el x 2^* + c 12 el x 3^* + c 13 el x 4^* + c 14 el x 5^* + c 15 el x 6^* + c 16 el x 7^* + c 17 el x 8^* + / * :

X J 1 col : Y J 2 col : XY X Y augment : J XY . 12 black plot : 41 line

J1084 mV 8.25 - 638 - range : i 1 mV rows range t i el mV i el J1084 eval : for mV t augment 31 line :

ΔJ1084 X Y Y X J1084 - (vectorize 3 round vectorize augment :

ΔJ1084 8.0957 - 210.002 - 0 7.7963 - 195.802 - 0 7.4807 - 182.962 - 0 3.7187 - 78.476 - 0 1.9063 - 38.862 - 0000 1.3739 26.87 0 5.2677 1000 6.4886 122.37 0 8.3743 156.634 0 12.5517 231.9681 0 14.7427 271.442 0.001 17.4928 321.108 0 17.8462 327.502 0 19.456 356.66 0 22.9259 419.58 0 24.3123 444.674 0 30.1095 548.23 0 34.9108 630.74 0 36.6933 660.37 0 55.669 961.93 0 61.716 1064.43 0 62.88 1084.5 0233 mat

Type 'J' ultimate fit to NBS 125 DONOT truncate

J J1084 2 1 sys

Calibrate the 'J' mV/I converter

XmV 100 - :

XmV °C ≡

mV 1 :

mV J1084 XmV - 0 ≡ mV 6 - 4 - solve mV 4.6322 -

4.6322 - J1084 10.7765 J1084 21 mat 100 - 20021 mat

XmV 200 :

XmV °C ≡

mV 1 :

mV J1084 XmV - mV 812 solve mV 10.7765

An industrial measuring T/C loop depends upon: 1. the supplier stated ±°C around the operating °C 2. added of the stated ±°C of the extension wiring. Their combination expressed by some common convention .. ............................

Example: 1. Operating => 200 °C 2. 'T' standard supply ± 1°C 3. Ext wiring ± 1°C 4. measured 200°C ± 1.5

TC ε 2^extension ε 2^+ sqrt

Tempting but ambiguous: T/C don't exhibit repeatable/fair manufacturing. On that fact, one may decide to adjust the XTR @ the 0.01 °C, that will partially correct @ operating.