Now that we have used either software compensation or hardware compensation to obtain a reference junction at 0°C, we must now convert the voltage measured V to temperature.

Unfortunately, the relationships between thermocouple voltage and temperature __are not linear.__

Figure 4. Thermocouple voltage as a function of temperature

Type | Metal A (+) | Metal B (-) |

E | Chromel | Constantan |

J | Iron | Constantan |

K | Chromel | Alumel |

R | Platinum | Platinum 13% Rhodium |

S | Platinum | Platinum 10% Rhodium |

T | Copper | Constantan |

In order to obtain a better picture of this non-linearity, let’s look at the **Seebeck coefficient as a function of temperature:**

Figure 5. Seebeck coefficient as a function of temperature for different types of thermocouples

Note that the type K thermocouple has a section which is almost linear between 0°C et 1000°C with a Seebeck coefficient α fluctuating around 40 µV/°C. Therefore this type of thermocouple can be used directly with a voltmeter multiplier and reference 0°C to obtain the temperature with moderate accuracy.

**Calculation using tables**

After reading the voltage value V, for example 8.35687 mV, with a type K thermocouple (Chromel/Alumel), let’s look in the ITS-90 table:

ITS-90 table for Type K ThermocoupleThermoelectric voltage in mV | |||||||||||

°C | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

190 | 7.739 | 7.779 | 7.819 | 7.859 | 7.899 | 7.939 | 7.979 | 8.019 | 8.059 | 8.099 | 8.138 |

200 | 8.138 | 8.178 | 8.218 | 8.258 | 8.298 | 8.338 | 8.378 | 8.418 | 8.458 | 8.499 | 8.539 |

210 | 8.539 | 8.579 | 8.619 | 8.659 | 8.699 | 8.739 | 8.779 | 8.819 | 8.860 | 8.900 | 8.940 |

We can see that this value is between Tinf 205 °C (8.338 mV) and Tsup 206 °C (8.378 mV). Let’s make a calculation by interpolation between the values 205 and 206 °C:

8.35687 – 8.338 = 0.01887 mV (remaining voltage above 205 °C)

8.378 – 8.338 = 0.040 mV for a difference of 1 °C

0.01887 / 0.040 = 0.471 °C in addition

The temperature is therefore 205 + 0.471 = **205.471 °C**

In summary, the equation is:

= 205 + [(8.35687 – 8.338) / (8.378 – 8.338)] = 205.471 °C

Or:

$$ T\left( {^\circ C} \right) = {T_{inf}}\left( {^\circ C} \right) + \frac{{V – {V_{inf}}}}{{{V_{sup}} – {V_{inf}}}} $$

**Calculation by polynomial equation**

It is possible to calculate the temperature from the thermoelectric voltage by means of a polynomial equation:

$$ {T_{90}} = {c_0} + {c_1}V + {c_2}{V^2} + \cdots + {c_n}{V^n} $$

T90 = Temperature in °C

V = Thermoelectric voltage in mV

c = Polynomial coefficients

n = Maximum order of the polynomial equation

Example of coefficients for type K thermocouples | |||

Temperature (°C) | -200 to 0 | 0 to 500 | 500 to 1372 |

Voltage (mV) | -5.891 to 0.000 | 0.000 to 20.644 | 20.644 to 54.886 |

c_{0} | 0 | 0.000000E+00 | -1.318058E+02 |

c_{1} | 2.5173462E+01 | 2.508355E+01 | 4.830222E+01 |

c_{2} | -1.1662878E+00 | 7.860106E-02 | -1.646031E+00 |

c_{3} | -1.0833638E+00 | -2.503131E-01 | 5.464731E-02 |

c_{4} | -8.9773540E-01 | 8.315270E-02 | -9.650715E-04 |

c_{5} | -3.7342377E-01 | -1.228034E-02 | 8.802193E-06 |

c_{6} | -8.6632643E-02 | 9.804036E-04 | -3.110810E-08 |

c_{7} | -1.0450598E-02 | -4.413030E-05 | 0 |

c_{8} | -5.1920577E-04 | 1.057734E-06 | 0 |

c_{9} | 0 | -1.052755E-08 | 0 |

Error (°C) | -0.02 to 0.04 | -0.05 to 0.04 | -0.05 to 0.06 |

The coefficients for other types of thermocouples are given in appendices at the end of the ITS-90 tables |