NIST ITS-90 conversion for every standard type — both directions, with cold-junction compensation.

How the NIST polynomials work

Every standard thermocouple type is defined by ITS-90 reference functions: the EMF (referenced to a 0 °C cold junction) is a polynomial in millivolts:

E (T) = i \sum c_{i} T^{i}

with coefficients published by NIST per type and temperature segment. Type K adds an exponential correction term $a_{0} e^{a_{1} (T - a_{2})^{2}}$ above 0 °C. The reverse direction uses separate inverse polynomials

T (E) = i \sum d_{i} E^{i}

valid only inside fixed EMF windows and accurate to roughly ±0.05 K. Evaluate them with Horner's method — never with naive pow() calls:

double poly(const double *c, int n, double x)
{
    double y = c[n - 1];
    for (int i = n - 2; i >= 0; i--)
        y = y * x + c[i];      /* Horner: one mul + one add per term */
    return y;
}

Cold-junction compensation works in EMF space, not temperature space:

E_{total} = E_{measured} + E (T_{cj})

then convert $E_{total}$ to temperature. Adding temperatures directly is wrong because the curve is non-linear.

Fixed-point implementation on an MCU

On a Cortex-M0 or similar FPU-less core, double-precision polynomials are slow and pull in soft-float code. The pragmatic embedded approach is a lookup table in integer µV with linear interpolation — deterministic, fast, and accurate to the table pitch (10 °C steps keep the interpolation error well under 0.1 °C for type K):

/* Type K, 0…1370 °C in 10 °C steps, EMF in µV (NIST table values) */
static const int32_t k_uV[138] = { 0, 397, 798, 1203, 1612, /* … */ };

int32_t k_uV_to_dC(int32_t uV)          /* returns 0.1 °C units */
{
    int lo = 0, hi = 137;
    while (hi - lo > 1) {               /* binary search the segment */
        int mid = (lo + hi) / 2;
        if (k_uV[mid] <= uV) lo = mid; else hi = mid;
    }
    /* linear interpolation inside the 10 °C segment */
    return lo * 100 + (int32_t)((int64_t)(uV - k_uV[lo]) * 100
                                / (k_uV[lo + 1] - k_uV[lo]));
}

Three details matter: keep the intermediate product in int64_t so it cannot overflow, return scaled integers (0.1 °C) instead of floats, and apply CJC by adding the cold-junction EMF (from a small second table around room temperature) before the lookup. If you prefer pure fixed-point polynomials, scale the coefficients to Q16.16 and evaluate with Horner using 64-bit accumulators.

Thermocouple Calculator

How the NIST polynomials work

Fixed-point implementation on an MCU