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03-03-99 The Callendar - van Dusen coef f icients.doc RS 1/3
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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
990303 / RS