NTC calibration and multiple temperature sensors

Introduction

Calibration of NTC sensors and different

NTC

Note that the temperature of the sensor rises when the current supplies through the resistor. The NTC is nonlinear; see below Calibration.

Negative Temperature Coefficient, NTCLE100E3101JB0 or similar (MF52B NTC Thermistor). The NTC is connected in series with a "shunt" resistor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_1} , see Measurement of resistors: voltage divider. Usually Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_1 = 10kOhms} is used. The total resistance of the circuit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = R_1 + R_{NTC}} , which gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} R_{NTC} = \frac{U_\text{measured}}{U-U_\text{measured}}R_1 \end{align} }

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} R_{NTC} = \frac{U-U_\text{measured}}{U_\text{measured}}R_1 \end{align} }

depending on the circuit. So check the circuit.

Calibrating: Steinhart-Hart Equation

A nonlinear Steinhart-Hart equation is widely used

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac 1T = A + B \ln(Rt) + C (\ln (Rt))^3 }

The parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} can be obtained, if the resistance of three (3) temperatures is known.

Calibrating using known datapoints

Though the NTC sensor is nonlinear, locally it will be linear. Thus by using some known datapoints the temperature can be estimated.

Some known datapoints:

• boiling water 100 deg
• room temperature
• freezing point of water

Semiconductor Physics

Resistivity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \rho J } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} is the density or number of charge carriers, $\rho =n/V$. Electrons are spin-1/2 particles and thus they obey Fermi-Dirac statistics. The electron current through a perpendicular semiconductor sample is

$J_{n}={\frac {I_{n}}{A}}=q\sum _{i=1}^{n}v_{i}=-qnv_{n}=qn\mu _{n}E$

and the total current is the sum of electrons and holes; $J=J_{n}+J_{p}=q(n\mu _{n}+p\mu _{p})E$. The proportionality constant is called conductivity, and its inverse is the resistivity:

$\rho ={\frac {1}{q(n\mu _{n}+p\mu _{p})}}$

Usually only other ($\mu _{n}$ or $\mu _{p}$) is dominant.

Obs! The term $\mu _{n}$ is called electron mobility and is defined by Newton II law:

${\begin{aligned}m_{n}v_{n}&=-qE\tau _{c}\\v_{n}&=-{\frac {q\tau _{c}}{m_{n}}}E\equiv -\mu _{n}E\end{aligned}}$

The Fermi energy is the highest energy of the collection of electrons at T=0 Kelvin is the "primitive" approximation. Fermi-Dirac distribution $f_{f-d}(\epsilon )={\frac {1}{e^{\beta (\epsilon -\mu )}+1}}$ can be approximated (exercise: when) as Maxwell-Boltzmann exponential $f_{M-B}(\epsilon )=e^{-\beta (\epsilon -\mu )}$