NTC calibration and multiple temperature sensors

Introduction

Calibration of NTC sensors and different

NTC thermistor elements come in many styles [4] such as axial-leaded glass-encapsulated (DO-35, DO-34 and DO-41 diodes), glass-coated chips, epoxy-coated with bare or insulated lead wire and surface-mount, as well as thin film versions. (Wikipedia)

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 $R_{1}$, see Measurement of resistors: voltage divider. Usually $R_{1}=10kOhms$ is used. The total resistance of the circuit is $R=R_{1}+R_{NTC}$, which gives

${\begin{aligned}R_{NTC}={\frac {U_{\text{measured}}}{U-U_{\text{measured}}}}R_{1}\end{aligned}}$

or

${\begin{aligned}R_{NTC}={\frac {U-U_{\text{measured}}}{U_{\text{measured}}}}R_{1}\end{aligned}}$

depending on the circuit. So check the circuit.

Calibrating: Steinhart-Hart Equation

A nonlinear Steinhart-Hart equation is widely used

${\frac {1}{T}}=A+B\ln(Rt)+C(\ln(Rt))^{3}$

The parameters $A$, 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

Some 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, 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 = n/V} . Electrons are spin-1/2 particles and thus they obey Fermi-Dirac statistics. The electron current through a perpendicular semiconductor sample 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 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; 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 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:

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 = \frac1{ q (n\mu_n + p\mu_p) } }

Usually only other (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 \mu_n} 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 \mu_p} ) is dominant.

Obs! The term 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 \mu_n} is called electron mobility and is defined by Newton II law:

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} m_n v_n &= - q E \tau_c \\ v_n &= -\frac{q\tau_c}{m_n}E \equiv - \mu_n E \end{align} }

The Fermi energy is the highest energy of the collection of electrons at T=0 Kelvin is the "primitive" approximation. Fermi-Dirac distribution 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 f_{f-d}(\epsilon) = \frac1{e^{\beta(\epsilon-\mu)}+1} } can be approximated (exercise: when) as Maxwell-Boltzmann exponential 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 f_{M-B}(\epsilon)=e^{-\beta(\epsilon-\mu)} }

With NTC thermistors, resistance decreases as temperature rises; usually due to an increase in conduction electrons bumped up by thermal agitation from the valence band (Wikipedia).