Calculation of temperature from PRTD resistance

I recently contributed a design for a simple platinum resistance temperature detector (PRTD) resistance two-wire 4-20mA transmitter circuit illustrated in Figure 1.

Figure 1 A two-wire, 4 to 20 mA current loop PRTD transmitter with 500 µA constant current sensor excitation. R1 and R2 are 0.1% tolerance, voltage reference is a 2.5-V LM4040x25

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Analog to digital conversion of the 4 to 20 mA Io reading is likewise simple and straightforward (a 250-Ω shunt resistor at the input of a 0 to 5-V ADC of adequate resolution and precision will do nicely) and getting from there to Rprtd is an easy chore in software (Io in milliamps):

PRTD resistance = R1(Io/Ix – 1) = 20Io – 10

The final step from there to a linear temperature measurement would be almost equally easy, thanks to Callendar Van Dusen (CVD) math, except for one annoying detail. The famous CVD polynomial is arranged to calculate PRTD resistance from temperature. Unfortunately, what we need is temperature from resistance!

Fortunately, another classic algebraic expression can ride to our rescue: The Quadratic Formula (QF).

Mixing vigorously CVD and QF and defining two constants:

u = 0.0039083 RPRTD@0oC

w = -0.0000005775 RPRTD@0oC

and one new variable,

x = RPRTD@0oC – PRTD

leads to a straightforward polynomial that will directly calculate a PRTD temperature from PRTD resistance that’s linear to within ±0.05oC over a temperature range spanning -80oC to +850oC.

 ToC= (-u + (u2 – 4wx)1/2)/(2w)

Stephen Woodward’s relationship with EDN’s DI column goes back quite a long way. Over 100 submissions have been accepted since his first contribution back in 1974.

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