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

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:

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