Infrared Temperature Calculation

An infrared sensor determines the temperature of an object by measuring the infrared radiation—or radiative flux—emitted from its surface. The radiative energy is a function of the object’s temperature and its material-specific emissivity. In addition, the thermal sensor also detects other infrared flux arriving, which includes reflected ambient radiation and internal radiation, as well as transmitted radiation. To convert the measured radiation into a temperature value, the infrared sensor uses a calibration model based on Planck’s law and incorporates the known or estimated emissivity of the target material, and in more complicated cases, the information about optical circumstances.

In the simplest case, only considering one radiation source and a narrowband wavelength, the Planck law has been defined for the spectral radiant exitance [math]M_\lambda^° (\lambda,T)[/math] of a grey body with the emissivity [math]\varepsilon[/math] on its temperature and wavelength, whereas [math]c_1[/math] and [math]c_2[/math] are Planck’s first and second constants.

[math]M_\lambda^° (λ,T)=ε \cdot \frac{c_1}{λ^5} \frac{1}{e^{\frac{c_2}{λ \cdot T}} -1} [/math]

The inversion of the equation provides the temperature for a given spectral exitance, including the emissivity.

[math]T=\frac{c_{2}}{\lambda \cdot \ln\left(\frac{\epsilon \cdot c_{1}}{\lambda^{5} \cdot M_\lambda^°}\right)+1}[/math]

In practice, detectors often measure radiation over a spectral range rather than at a single wavelength. To account for this, the cumulative intensity is obtained by integrating the Planck function over the desired wavelength interval, from the lower to the upper wavelength bound.

[math]\phi_{\lambda}°\left(\lambda,T\right)=\int_{\lambda_{1}}^{\lambda_{2}} \frac{c_{1}}{\lambda^{5}}\frac{1}{e^{\frac{c_{2}}{\lambda\cdot T}}-1}d\lambda[/math]

No closed-form solution exists for temperature in this equation, so the integral must be evaluated numerically—typically using methods such as polynomial approximations. To simplify, the assumption is that the detector measures the received infrared flux [math]\phi[/math] in comparison to the sensor temperature. This total flux [math]\phi_{\lambda}[/math] is proportional to the signal detected U by the infrared sensor and forms the basis for determining the object temperature. The self-radiation of the infrared thermometer must be taken into account. Therefore, the temperature of the sensor itself [math]T_{int}[/math] must be measured and subtracted from the overall amount of infrared flux later.

[math]U\ \sim \ \phi_\lambda\left(\ \lambda,\ T\right)-\ \phi_{int}\left(T_{int}\right)\ [/math]

Based on the Kirchhoff law and the Lambert law, the radiation flux [math]\phi_{\lambda}[/math] is the sum of the emitted radiation, reflected radiation, and transmitted radiation, whereas, [math]\theta[/math] is the angle between the surface normal and the radiation direction, [math]\phi[/math] is the angle of direction in the half-space, and the term [math]cos\left(\theta\right)[/math] accounts for Lambert’s cosine law.

[math]\phi=\phi_{\epsilon}+\phi_{p}+\phi_{\tau}=\sum_i\int_{\lambda_{1}}^{\lambda_{2}} M_{i}\left(\lambda,T\right)d\lambda[/math]

The radiation flux [math]\phi_{\lambda}[/math], can be derived by integrating Planck’s law over the bandwidth of the infrared sensor from all infrared radiation sources and considering Lambert’s Law. Here, the assumption is that the emissivity is approximately constant over the range and the sensor sensitivity is assumed to be 1, the equation simplifies to the following, whereas the variable [math]T_{obj}[/math], [math]T_{p}[/math], [math]T_{\tau}[/math] represent the temperature of the object, the temperature of a reflecting source, and the temperature of the background.

[math]\phi_{\lambda}\left(\lambda,T_{obj}T_{p},T_{\tau}\right)=\int_{\lambda_{1}}^{\lambda_{2}} \epsilon \cdot M°\left(\lambda,T_{obj}\right)d\lambda+\int_{\lambda_{1}}^{\lambda_{2}}\rho \cdot  M°\left(\lambda,T_{\rho}\right)d\lambda+\int_{\lambda_{1}}^{\lambda_{2}}\tau \cdot M°\left(\lambda,T_{\tau}\right)d\lambda[/math]

Therefore, the proportional detector signal U is a combination of all radiation flux.

[math]U\sim \phi_{\lambda}\left(\lambda,T_{obj}T_{\rho},T_{\tau}\right)-\phi_{int}\left(T_{int}\right)[/math]

As a simplification, according to the Stefan-Boltzmann law, the radiated energy from a surface is proportional to its emissivity and the fourth power of its absolute temperature, for the total infrared spectrum [math]\lambda_1=0[/math] and [math]\lambda_2=\infty [/math].

[math]U\sim \phi\left(T\right)=\epsilon \cdot \sigma\cdot T_{4}[/math]

Since practical infrared thermometers operate over a limited spectral range, the ideal Stefan-Boltzmann exponent n=4 is replaced by an effective exponent n, that depends on the wavelength and spectral sensitivity. The constant c_i is introduced here to replace optical parameters and acts as calibration coefficient.

[math]U\sim \phi_{\lambda}\left(\lambda,T\right)\sim c_{i}T^{n}[/math]

The detector signal is based on the total radiation flux, which is the sum of reflections, emission, and transmissivity, and subtracted by the sensor’s thermal emission.

[math]U\ \sim \ \phi_\lambda\left(\ \lambda,T_{obj}T_\rho,\ T_\tau\right)-\ \phi_{int}\left(T_{int}\right)\rightarrow U=c_\varepsilon\ \varepsilon\ \ T_{obj}^n+c_\rho\ \rho\ \ T_{\rho\ }^n+\ c_\tau\ \tau\ \ T_{\tau\ }^n-\ c_{int}\ T_{int}^n[/math]

By solving the inverse radiation equation, the sensor translates the measured infrared signal into a temperature reading that corresponds to the thermal state of the object’s surface.

[math]T_{obj}=\ \sqrt[n]{\frac{U+ c_{int} T_{int}^n{- c}_ρ ρ  T_{ρ }^n-c_τ τ  T_{τ }^n }{c_ε \cdot ε}}[/math]

This formulation represents an idealized measurement setup without any window in the optical path.

Figure 2 represents a standard measurement setup, as a representative example. An opaque object with negligible transmissivity, directly viewed by the non-contact thermal sensor. In this scenario, no transmissive elements are present between the object and the sensor. Additionally, the object is of high emissivity and optically thick, allowing us to assume zero transmissivity of the object [math]\tau=0[/math]. Due to the Kirchhoff law, we derive that [math]\rho=1- \ \varepsilon[/math]. The thermal equilibrium between the sensor and its environment. Therefore, we can assume that the ambient temperature is equal to the sensor temperature [math]T_{int}=T_{amb}[/math]. Additionally, it is assumed that the temperature of any reflection is equal to the ambient temperature [math]T_\rho=T_{amb}[/math]. Therefore, the equation simplifies to the following.

[math]T_{obj}=\ \sqrt[n]{\frac{U+T_{int}^n\ \left(c_{int}+\ c_\rho\ \varepsilon\ -\ c_\rho\right)\ }{c_\varepsilon\cdot\ \varepsilon}}[/math]

Accurate temperature determination thus requires either prior knowledge of the emissivity or an in-situ method to compensate for it. The calibration coefficient [math]c_{int}[/math], [math]c_{\rho}[/math], [math]c_{\epsilon}[/math] are determined during the factory calibration.

A more complex scenario involves incorporating an infrared window between the object and the sensor, as illustrated in Figure 3. The object emits infrared radiation corresponding to its temperature while also reflecting some ambient radiation. The sensor measures the total radiative flux passing through the window, which has a transmissivity of [math]\tau_w[/math]. The window partially absorbs the incoming radiation and emits additional radiation depending on its temperature, [math]T_{window}[/math].

The sum of the radiation flux changes to the following equation.

[math]\phi=\ \tau_w\ \phi_\varepsilon+\ \tau_w\ \phi_\rho+\ \tau_w\ \phi_{\tau\ }+\ \phi_{window\ \varepsilon}+\phi_{window\ \rho} \\ = \tau_w \cdot\sum_{i}\int_{\lambda_1}^{\lambda_2}{M_i\left(\lambda,\ T\right)\ d\lambda}+\sum_{i}\int_{\lambda_1}^{\lambda_2}{M_{window\ i}\left(\lambda,\ T_{window}\right)\ d\lambda}[/math]

In most cases the window is positioned very close to the infrared sensor, so that the assumption can be made that the window has the same temperature as the ambient which is equal to the temperature of the sensor, [math]\ T_{window}=\ \ T_{amb}=\ \ T_{int}[/math]. Considering the same assumptions regarding the Planck curve, Stefan-Boltzmann law, calibration coefficient, and self-radiation of the device as outlined in detail before, the detector signal equals the following equation.

[math]U\ \sim \ \phi_\lambda\left(\ \lambda,T_{obj}T_{int}\right)-\ \phi_{int}\left(T_{int}\right)\rightarrow U \\ =\tau_w\left(\ c_\varepsilon\ \varepsilon\ \ T_{obj}^n\ +c_\rho\ \rho\ \ T_{int\ }^n+\ c_\tau\ \tau\ \ T_{back\ }^n\right)+\left(\ c_\varepsilon\ \left(\varepsilon_w+\ \rho_w\right)\ -\ c_{int}\right)\ T_{int}^n[/math]

Similar to before, the equation is rearranged to solve for the object temperature.

[math]T_{obj}=\ \sqrt[n]{\frac{U+T_{int}^n\ \left[c_{int}-\ \tau_wc_\rho\ \rho-c_\varepsilon\left(\varepsilon_w+\ \rho_w\right)\ \right]-\ \tau_w\ c_\tau\ \tau\ \ T_\tau^n\ }{\tau_w\cdot \ c_\varepsilon\cdot\ \varepsilon}}[/math]

Considering that most infrared windows have high transmissivity and low emissivity anyway, the assumption [math]\varepsilon_w[/math],[math]\ \rho_w\approx0[/math]. Applying Kirchhoff’s law, we derive that [math]\rho=1- \ \varepsilon[/math], and assuming that the temperature in the back of the object is the same as the ambient temperature [math]T_{\tau}=T_{amb}=T_{int}[/math] , the following equation is concluded.

[math]T_{obj}=\ \sqrt[n]{\frac{U+T_{int}^n\ -\left[c_{int}+\tau_w\left(c_\rho\varepsilon-\ c_\rho-\ \ c_\tau\ \tau\right)\ \right]\ }{\tau_w\cdot \ c_\varepsilon\cdot\ \varepsilon}}[/math]