A typical thermocouple circuit can be illustrated as follows:
Typical Thermocouple Circuit
Suppose that the Seebeck coefficients of two dissimilar metallic materials, metal A and metal B, and the lead wires are S_{A}, S_{B}, and S_{Lead} respectively. All three Seebeck coefficients are functions of temperature. The voltage output V_{out} measured at the gage (see schematic above) is,
where T_{Ref} is the temperature at the reference point, T_{Tip} is the temperature at the probe tip. Note that mathematically the voltage induced by the temperature and/or material mismatch of the lead wires cancels, whereas in reality the lead wires will introduce noise into the circuit.
If the Seebeck coefficient functions of the two thermocouple wire materials are precalibrated and the reference temperature T_{Ref} is known (usually set by a 0°C ice bath), the temperature at the probe tip becomes the only unknown and can be directly related to the voltage readout.
If the Seebeck coefficients are nearly constant across the targeted temperature range, the integral in the above equation can be simplified, allowing one to solve directly for the temperature at the probe tip,
In practice, vendors will provide calibration functions for their products. These functions are usually high order polynomials and are calibrated with respect to a certain reference temperature, e.g., 0 °C (32 °F). Suppose that the coefficients of the calibration polynomials are a_{0}, a_{1}, a_{2}, ..., a_{n}. The temperature at the probe tip can then be related to the voltage output as,
Note that the above formula is effective only if the reference temperature T_{Ref} in the experiment is kept the same as the reference temperature specified on the data sheet. Furthermore, these coefficients are unit sensitive. Make sure to use the vendorspecified temperature unit (i.e. Celsius/centigrade, Fahrenheit, or Kelvin) when plugging in numbers.
Again, a thermocouple is a relative not absolute temperature sensor. In other words, a thermocouple requires a reference of known temperature which is provided by ice water in the above illustration. While ice water is an easy to obtain and well known reference, it is not practical out side of a laboratory. Thus, common commercialized thermocouples often includes another temperature sensor, such as thermistor, to provide the reading of the reference (room/surrounding) temperature.
