# Type K Thermocouple CalibrationEfficient voltage to temperature conversion and temperature measurement using rational polynomial functions

Mosaic's Thermocouple Wildcard automatically converts measured voltages to temperatures for you, and with high accuracy.

However, if you are not using the Mosaic Thermocouple Wildcard, but looking for accurate equations for type K thermocouple measurement, we hope the type K thermocouple calibration coefficients and methods here are helpful.

Equations for voltage to temperature conversion usually use high order polynomials. NIST, the National Institute of Standards and Technology, provides a database of thermocouple values and a high order polynomial formula curve fitted to type K thermocouple data.

However, the polynomial equations NIST provides aren't as accurate as they should be – they produce errors of fit as the high-order polynomial functions oscillate around the data, as shown in these curve fits of NIST polynomials to Type K thermocouple data.

Fortunately, there are more accurate ways to fit the data – using rational polynomial functions. This page shows you how to convert type K thermocouple voltage to temperature with rational polynomial functions accurately and efficiently – more so than you can using NIST's higher order polynomials.

## Type K calibration equation

A rational polynomial function approximation for Type K 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 National Institute of Standards and Technology (NIST) type K 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 K 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 K type thermocouple data.

The following table contains calibration coefficients for Type K thermocouple wire.

 Range Voltage: Temperature: Coefficients T0 -6.404 to -3.554 mV -3.554 to 4.096 mV 4.096 to 16.397 mV 16.397 to 33.275 mV 33.275 to 69.553 mV -250 to -100°C -100 to 100°C 100 to 400°C 400 to 800°C 800 to 1200°C `-1.2147164E+02` `-8.7935962E+00` `3.1018976E+02` `6.0572562E+02` `1.0184705E+03` `-4.1790858E+00` `-3.4489914E-01` `1.2631386E+01` `2.5148718E+01` `4.1993851E+01` `3.6069513E+01` `2.5678719E+01` `2.4061949E+01` `2.3539401E+01` `2.5783239E+01` `3.0722076E+01` `-4.9887904E-01` `4.0158622E+00` `4.6547228E-02` `-1.8363403E+00` `7.7913860E+00` `-4.4705222E-01` `2.6853917E-01` `1.3444400E-02` `5.6176662E-02` `5.2593991E-01` `-4.4869203E-02` `-9.7188544E-03` `5.9236853E-04` `1.8532400E-04` `9.3939547E-01` `2.3893439E-04` `1.6995872E-01` `8.3445513E-04` `-7.4803355E-02` `2.7791285E-01` `-2.0397750E-02` `1.1413069E-02` `4.6121445E-04` `2.3841860E-03` `2.5163349E-02` `-1.8424107E-03` `-3.9275155E-04` `2.5488122E-05` `0.0`

## Type K thermocouple accuracy and calibration error

The following graph shows residual errors after calibrating the thermocouple with a rational function of polynomials. Fig. 2  K type thermocouple residual temperature errors after calibrating the thermocouple with a rational function of polynomials.

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 K cold junction voltages

Your type K 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 K 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:

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