# Type E Thermocouple Calibration

For those using Mosaic's Thermocouple Wildcard, thermocouple voltage measurements are converted to temperatures for you by the pre-coded software drivers, and with high accuracy.

But if you are not using the Mosaic Thermocouple Wildcard, and need accurate equations for thermocouple voltage to temperature conversion, we hope the thermocouple calibration equations and coefficients provided here are helpful.

Equations for voltage-temperature conversion are usually based on high-order polynomials. In fact, the National Institute of Standards and Technology (NIST database of thermocouple values) provides a tenth-order polynomial formula along with coefficients curve fitted to their thermocouple data.

However, even very high order polynomials are often not very accurate – they produce systematic errors as the polynomial functions ripple around the desired data, as shown in these curve fits to Type K thermocouple data.

But there's a better way – rational polynomial functions. This page shows you how to convert thermocouple voltage-to-temperature^{1)} with rational polynomial functions more accurately, and more efficiently, than you can with the more commonly used higher order polynomials.

The rational polynomial coefficients provided here produce an order of magnitude lower errors than the NIST ITS-90 thermocouple coefficients for direct and inverse polynomials.

## Type E calibration equation

A rational polynomial function for Type E thermocouples is used for computing temperature from measured thermocouple potential. The calibration equation uses a ratio of two lower order polynomials rather than one high order polynomial. Using a least squares curve fitting procedure we fit the National Institute of Standards and Technology (NIST) ITS-90 type E 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 *T _{o}*,

*V*, and the

_{o}*p*and

_{i}*q*are coefficients. The function uses a ratio of two polynomials,

_{i}*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, *T _{o}*,

*V*,

_{o}*p*and

_{i}*q*were found by performing a least squares curve fit to the NIST data for type E 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 measurement data.

_{i}The following table contains calibration coefficients for Type E thermocouple wire.

Range | |||||
---|---|---|---|---|---|

Voltage: | -9.835 to -5.237 mV | -5.237 to 0.591 mV | 0.591 to 24.964 mV | 24.964 to 53.112 mV | 53.112 to 76.373 mV |

Temperature: | -270 to -100°C | -100 to 10'°C | 10 to 350°C | 350 to 700°C | 700 to 1000°C |

Coefficients | |||||

T_{0} | `-1.1721668E+02` | `-5.0000000E+01` | `2.5014600E+02` | `6.0139890E+02` | `8.0435911E+02` |

V_{0} | `-5.9901698E+00` | `-2.7871777E+00` | `1.7191713E+01` | `4.5206167E+01` | `6.1359178E+01` |

p_{1} | `2.3647275E+01` | `1.9022736E+01` | `1.3115522E+01` | `1.2399357E+01` | `1.2759508E+01` |

p_{2} | `1.2807377E+01` | `-1.7042725E+00` | `1.1780364E+00` | `4.3399963E-01` | `-1.1116072E+00` |

p_{3} | `2.0665069E+00` | `-3.5195189E-01` | `3.6422433E-02` | `9.1967085E-03` | `3.5332536E-02` |

p_{4} | `8.6513472E-02` | `4.7766102E-03` | `3.9584261E-04` | `1.6901585E-04` | `3.3080380E-05` |

q_{1} | `5.8995860E-01` | `-6.5379760E-02` | `9.3112756E-02` | `3.4424680E-02` | `-8.8196889E-02` |

q_{2} | `1.0960713E-01` | `-2.1732833E-02` | `2.9804232E-03` | `6.9741215E-04` | `2.8497415E-03` |

q_{3} | `6.1769588E-03` | `0.0` | `3.3263032E-05` | `1.2946992E-05` | `0.0` |

## Type E thermocouple accuracy and calibration error

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. 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 E cold junction voltages

Your type E 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 E 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 *T _{CJ}* is the cold junction temperature,

*V*is the computed cold junction voltage, and the

_{CJ}*T*,

_{0}*V*,

_{0}*p*and

_{i}*q*are coefficients.

_{i}To learn more about cold junction compensation you can consult How To Do Cold Junction Compensation, and you can find the coefficients here: