# Type N Thermocouple Calibration

Convert type N thermocouple voltages to temperatures efficiently using rational polynomial function calibration equations.

If you use Mosaic's Thermocouple Wildcard with type N thermocouples, voltage measurements are automatically converted to temperatures by the included software, and with high accuracy.

If you are not using the Wildcard, but you are looking for accurate equations for type N thermocouple measurement, we hope the thermocouple calibration equations and coefficients we provide here are helpful.

Equations for converting type N voltage to temperature usually are based on high-order polynomials. In fact, the NIST database of thermocouple values provides a tenth order polynomial formula with coefficients curve fitted to type N thermocouple data.

However, even very high order polynomials are not very accurate as the polynomial functions oscillate around the measurement data, as shown in curve fits to Type K thermocouple data.

Fortunately, there's a better way – you can use rational polynomial functions. This page shows you how to convert type N thermocouple voltage to temperature with ratios of lower order polynomial functions more accurately and more efficiently than you could with the more commonly used single higher order polynomials.

The type N rational polynomial coefficients provided here produce much lower errors than the NIST ITS-90 type N thermocouple coefficients for direct and inverse polynomials.

## Type N calibration equation

A rational polynomial function approximation for Type N 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 N 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 N 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 type N thermocouple calibration data.

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

 Range Voltage: Temperature: -4.313 to 0 mV 0 to 20.613 mV 20.613 to 47.513 mV -250 to 0°C 0 to 600°C 600 to 1300°C `-5.9610511E+01` `3.1534505E+02` `1.0340172E+03` `-1.5000000E+00` `9.8870997E+00` `3.7565475E+01` `4.2021322E+01` `2.7988676E+01` `2.6029492E+01` `4.7244037E+00` `1.5417343E+00` `-6.0783095E-01` `-6.1153213E+00` `-1.4689457E-01` `-9.7742562E-03` `-9.9980337E-01` `-6.8322712E-03` `-3.3148813E-06` `1.6385664E-01` `6.2600036E-02` `-2.5351881E-02` `-1.4994026E-01` `-5.1489572E-03` `-3.8746827E-04` `-3.0810372E-02` `-2.8835863E-04` `1.7088177E-06`

## Type N thermocouple accuracy and calibration error

The following graph shows residual 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 N cold junction voltages

Your type N 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 N 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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