Type R Thermocouple Calibration
Mosaic's Thermocouple Wildcard incorporates high accuracy thermocouple voltage to temperature conversion. For those of you not using the Thermocouple Wildcard, but looking for highly accurate equations for type R thermocouple measurement, we hope the calibration equations and coefficients here are helpful.
Equations for type R voltage to temperature conversion are usually based on high-order polynomials. In fact, the ITS-90 NIST database of thermocouple values provides a high order polynomial formula along with coefficients curve fitted to type R thermocouple data.
Unfortunately the high order polynomials used introduce errors as the polynomial functions oscillate around the data, as you can see in these curve fits to Type K thermocouple data.
But there is a better way to do it – you can use ratios of lower order polynomial functions to fit the data more accurately. This page shows you how to convert type R thermocouple voltages to temperatures using rational polynomial functions more accurately and efficiently than you can with higher order polynomials.
Type R calibration equation
A rational polynomial function approximation for Type R thermocouples is used for computing temperature from measured thermocouple potential. The calibration equation uses a ratio of two smaller order polynomials rather than one large order polynomial. Using a least squares curve fitting procedure we fit the NIST type R thermocouple data with a rational function of the following form,
where T is the thermocouple temperature (in °C), V is the thermocouple voltage (in millivolts), and To, Vo, and the pi and qi are coefficients. The function uses a ratio of two polynomials, P/Q, in this case a fourth order to a third order polynomial. The second form of the equation emphasizes the most efficient order of operations.
The coefficients, To, Vo, pi and qi were found by performing a least squares curve fit to the NIST data for type R thermocouples. The full temperature range is broken into several sub-ranges, and a different set of coefficients used for each. The coefficients, fitted data and charts of residual errors may be found in this Excel spreadsheet of thermocouple data.
The following table contains calibration coefficients for Type R thermocouple wire. The graph shows residual errors after calibrating the thermocouple with a rational function of polynomials. Most of the residual error results from rounding of the NIST provided voltage values to the nearest microvolt. The rational function approximation interpolates the data well so that computed values of temperature are more accurate than the residuals plots would indicate.
The following table is of calibration coefficients for Type R thermocouple wires.
| Range | ||||
|---|---|---|---|---|
| Voltage: | -0.226 to 1.469 mV | 1.469 to 7.461 mV | 7.461 to 14.277 mV | 14.277 to 21.101 mV |
| Temperature: | -50 to 200°C | 200 to 760°C | 760 to 1275°C | 1275 to 1768°C |
| Coefficients | ||||
| T0 | 1.3054315E+02 | 5.4188181E+02 | 1.0382132E+03 | 1.5676133E+03 |
| V0 | 8.8333090E-01 | 4.9312886E+00 | 1.1014763E+01 | 1.8397910E+01 |
| p1 | 1.2557377E+02 | 9.0208190E+01 | 7.4669343E+01 | 7.1646299E+01 |
| p2 | 1.3900275E+02 | 6.1762254E+00 | 3.4090711E+00 | -1.0866763E+00 |
| p3 | 3.3035469E+01 | -1.2279323E+00 | -1.4511205E-01 | -2.0968371E+00 |
| p4 | -8.5195924E-01 | 1.4873153E-02 | 6.3077387E-03 | -7.6741168E-01 |
| q1 | 1.2232896E+00 | 8.7670455E-02 | 5.6880253E-02 | -1.9712341E-02 |
| q2 | 3.5603023E-01 | -1.2906694E-02 | -2.0512736E-03 | -2.9903595E-02 |
| q3 | 0.0 | 0.0 | 0.0 | -1.0766878E-02 |
Type R thermocouple accuracy and calibration error
Here's a graph that shows the errors remaining after calibration using rational polynomial functions.
Most of the residual error in the above graph results from rounding of the NIST provided voltage values to the nearest microvolt. You can see the rounding errors as aliasing or Moiré patterns in the data.
The rational function approximation interpolates the data well so that computed values of temperature are more accurate than the residuals plots would indicate.
Computing type R cold junction voltages
Your type R thermocouple may be terminated at a cold junction temperature other than 0°C. If so, you must measure the temperature of the cold junction using another sensor, perhaps a thermistor. You can then compute a cold junction voltage from that measured temperature, and use it to compensate the type R thermocouple voltage before converting the thermocouple voltage to a temperature.
In that case you need the inverse transform, that is, an equation for converting temperature into voltage. You don't need a valid equation for a wide temperature range as the cold junction is likely to be placed at ambient temperature. Usually a range of -20 to +70°C is sufficient.
To convert temperature to voltage, you can again use a rational function approximation of the form,
where TCJ is the cold junction temperature, VCJ is the computed cold junction voltage, and the T0, V0, pi and qi are coefficients.
To learn more about cold junction compensation you can consult How To Do Cold Junction Compensation, and you can find the coefficients here: