Converting thermocouple voltages to temperature.

Background

Unlike most sensors used in data acquisition, thermocouples do not have a linear mapping between their voltage and celcius values. This means that instead of factoring some scalar value against the data, in order to scale it to engineering units (eg. volts to acceleration), an experimentally derived equation must be created to approximate the temperature at specific voltages. This non-linearity is primarily due to the materials being used to discover the temperature difference.

The science and engineering behind thermocouples goes beyond the scope of this post but more information can be found below:

• Wikipedia: Thermoelectric Effect
• Wikipedia: Thermocouple
• Reference Temperatures
• Simple Demonstration of the Seebeck Effect

K-Type Thermocouples

One of the most general purpose thermocouple types used in data acquisition are K types since their are relatively inexpensive to make. This will also be the thermocouple that I will focus on.

Luckily, NIST provides a table of discrete values, correlating voltages to celcius and vice-versa. In addition, they also provide the exact polynomial equation to use underneath the given tables, depending on the voltage range the value falls in. Following is the exact functions to do the conversion for K-type thermocouples.

Voltage To Celcius

Given some ordered set of coefficients, D, and a voltages value, v, let the voltage to celcius conversion be defined as a function of v as follows.

D is dependent on the voltage value being evaluated:

• values greater than 206.44 mV will have D = {-131.8058, 48.30222, -1.646031, 0.05464731, -0.0009650715, 0.000008802193, -0.0000000311081, 0.0, 0.0, 0.0 };
• values geater than 0.0 V will have D = { 0.0, 25.08355, 0.07860106, -0.2503131, 0.0831527, -0.01228034, 0.0009804036, -0.0000441303, 0.000001057734,-0.00000001052755};
• and values less than 0.0 V will have D = { 0.0, 25.173462, -1.1662878, -1.0833638, -0.8977354, -0.37342377, -0.086632643, -0.010450598, -0.00051920577, 0.0 }.

Celcius To Voltage

Given some ordered set of coefficients, C, an ordered set of exponentials, A, and a celcius value, x, let the celcius to voltage conversion be defined as a function of x as follows.

C is dependent on the celius value being evaluated:

• positive and zero values will have C = { -0.017600413686, 0.038921204975, 0.000018558770032, -0.000000099457592874, 0.00000000031840945719, -0.00000000000056072844889, 0.00000000000000056075059059, -3.2020720003E-19, 9.7151147152E-23, -1.2104721275E-26};
• negative values will have C = { 0.0, 0.039450128025, 0.000023622373598, -0.00000032858906784, -0.0000000049904828777, -0.000000000067509059173, -0.00000000000057410327428, -0.0000000000000031088872894, -1.0451609365E-17, -1.9889266878E-20, -1.6322697486E-23};

A is a fixed ordered set of exponentials; A = {0.1185976, -0.0001183432, 126.9686}.

Performance

Since the polynomial equation's number of terms are bounded by a constant (ie. |D| or |C|) the performance hit will be a constant factor to the number of voltage or celcius values to calculate. Also, temperature is sampled at a very low rate, yielding small data sets that usually have redundant values in them. Further optimization can be made by caching previous computations and reusing their results if the same voltage or celcius value is encountered again.

The Code

private static double[] _thermocoupleCoefficientsTypeKneg = {
 0.0,
 0.039450128025,
 2.3622373598E-05,
 -3.2858906784E-07,
 -4.9904828777E-09,
 -6.7509059173E-11,
 -5.7410327428E-13,
 -3.1088872894E-15,
 -1.0451609365E-17,
 -1.9889266878E-20,
 -1.6322697486E-23
};
private static double[] _thermocoupleCoefficientsTypeKpos = {
 -0.017600413686,
 0.038921204975,
 1.8558770032E-05,
 -9.9457592874E-08,
 3.1840945719E-10,
 -5.6072844889E-13,
 5.6075059059E-16,
 -3.2020720003E-19,
 9.7151147152E-23,
 -1.2104721275E-26
};
private static double[] _thermocoupleExponentialsTypeK = {
 0.1185976,
 -0.0001183432,
 126.9686
};
/// 

/// Type K thermocouple inverse coefficient values for voltage to celcius conversions.
/// 
/// Valid values for -200C - 0C / -5.891mV - 0mV.
private static double[] _thermocoupleInverseCoefficientsTypeK0 = {
 0.0,
 25.173462,
 -1.1662878,
 -1.0833638,
 -0.8977354,
 -0.37342377,
 -0.086632643,
 -0.010450598,
 -0.00051920577,
 0.0
};
/// 

/// Type K thermocouple inverse coefficient values for voltage to celcius conversions.
/// 
/// Valid values for 0C - 500C / 0mV - 20.644mV.
private static double[] _thermocoupleInverseCoefficientsTypeK1 = {
 0.0,
 25.08355,
 0.07860106,
 -0.2503131,
 0.0831527,
 -0.01228034,
 0.0009804036,
 -4.41303E-05,
 1.057734E-06,
 -1.052755E-08
};
/// 

/// Type K thermocouple inverse coefficient values for voltage to celcius conversions.
/// 
/// Valid values for 500C - 1372C / 20.644mV - 54.886mV.
private static double[] _thermocoupleInverseCoefficientsTypeK2 = {
 -131.8058,
 48.30222,
 -1.646031,
 0.05464731,
 -0.0009650715,
 8.802193E-06,
 -3.11081E-08,
 0.0,
 0.0,
 0.0
};


public static double CelciusToVoltageTypeK(double value)
{
 double[] a = _thermocoupleExponentialsTypeK;
 double[] c = null;
 double result = 0.0;
 if ((value >= 0.0)) {
  c = _thermocoupleCoefficientsTypeKpos;
 } else {
  c = _thermocoupleCoefficientsTypeKneg;
 }
 for (int index = 0; index <= c.Length - 1; index++) {
  result += (c[index] * Math.Pow(value, index)) + (a[0] * Math.Exp(a[1] * Math.Pow(value - a[2], 2)));
 }
 return result;
}

public static double VoltageToCelciusTypeK(double value)
{
 double[] d = null;
 double result = 0.0;
 if ((value > 0.020644)) {
  d = _thermocoupleInverseCoefficientsTypeK2;
 } else if ((value >= 0.0)) {
  d = _thermocoupleInverseCoefficientsTypeK1;
 } else {
  d = _thermocoupleInverseCoefficientsTypeK0;
 }
 for (int index = 0; index <= d.Length - 1; index++) {
  result += d[index] * Math.Pow(value, index);
 }
 return result;
}