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0,00

5,00

10,00

15,00

20,00

25,00

30,00

35,00

-2

00

-1

00 0

1

00

2

00

3

00

4

00

5

00

6

00

7

00

8

00

The Callendar van Dusen coefficients

The platinum thermometer is one of the most linear and practical temperature transdu-

cers in existance. Yet it is still necessary to linearise the measured signal, as w ill appearfrom the diagram below . The diagram illustrates the disparity in ohms betw een the ac-

tual resistance value at a given temperature and the value that w ould be obtained by asimple linear calculation for a Pt100 sensor:

According to IEC751, the non-linearity of the platinum thermometer can be expressedas:

])100(CBA1[ 320 ttttRRt +++= (1)

in which C is only applicable when t < 0 C.

The coefficients A, B, and C for a standard sensor are stated in IEC751. If a standardsensor is not available or if a greater accuracy is required than can be obtained from the

coefficients in the standard, the coefficients can be measured individually for each sen-sor. This can be done e.g. by determining the resistance value at a number of known

temperatures and then determining the coefficients A, B, and C by regression analysis.

The Callendar van Dusen method:

However, an alternative method for determination of these coeff icients exists. Thismethod is based on the measuring of 4 known temperatures:

Measure R0 at t0 = 0 C (the freezing point of w ater)

Measure R100 at t100 = 100 C (t he boiling point of w at er)

Figur 1. Deviat ion in ohms betw een the actual resistance value

and the linear interpolation as a function of the temperatureexpressed in C.

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Measure Rh at th = a high temperature (e.g. the freezing point of zink, 419.53 C)

Measure Rl at t l: = a low temperature (e.g. the boiling point of oxygen, -182.96 C)

Calculation of :

First the linear parameter is determined as the normalised slope between 0 and 100 C:

0

0100

100 R

RR

= (2)

If this rough approximation is enough, the resistance at other temperatures can be cal-

culated as:

tRRRt += 00 (3)

and the temperature as a function of the resistance value as:

=

0

0

RRRt t (4)

Calculation of :

Callendar has established a better approximation by introducing a term of the second

order, , into the function. The calculation of is based on the disparity between the

actual temperature, th, and the temperature calculated in (4):

))(1( 10010 0

0

0

hh

hh

tt

R

RtRt

=

(5)

With the introduction of into the equation, the resistance value for positive temperatu-

res can be calculated w ith great accuracy:

][ ))(1(100100

00

tttRRRt += (6)

Calculation of :

At negative temperatures (6) w ill still give a small deviat ion as shown in figure 1 (bot-

tom curve). Van Dusen therefore introduced a term of the fourth order, , which is only

applicable for t< 0 C. The calculation of is based on the disparity between the actualtemperature, t l, and the temperature that would result from employing only and :

3

0

0

))(1(

))(1(

100100

10010 0][

ll

ll

tt

tt

R

RtR llt

+

=

(7)

With the introduction of both Callendar' s and van Dusen' s constant, the resistance va-

lue can be calculated correctly for the entire temperature range, as long as one remem-

bers to set = 0 for t> 0 C:

][3

00 ))(1())(1(10010 0100100

tttttRRRt += (8)

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Conversion to A, B, and C:

Equation (8) is the necessary tool for accurate temperature determination. However,

seeing that the IEC751 coeff icients A, B, and C are more w idely used, it w ould be natu-

ral to convert to these coefficients:Equation (1) can be expanded to:

)CC100BA1( 4320 ttttRRt +++= (9)

and by simple coefficient comparison with equation (8) the follow ing can be determined:

100A

+= (10)

2100B

= (11)

4100C

= (12)

As an example, t he table below shows both sets of coeff icients f or a Pt100 resistoraccording to the IEC751 and ITS90 scale:

0,003850 A 3,908 x 10 -3

1,4999 B -5,775 x 10 -7

0,10863 C -4,183 x 10 -12

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