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.

B equation

Steinhart--Hart equation can be derived from extending the B parameter equation to an infinite series.

$R=R_{0}e^{B({\frac {1}{T}}-{\frac {1}{T_{0}}})}$

Steinhart-Hart Equation

A nonlinear Steinhart-Hart equation is widely used

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

The parameters $A$, $B$ and $C$ can be obtained, if the resistance of three (3) temperatures is known.

Steinhart-Hart Equation was found in 1968.

$\rho ^{-1}=F(T)e^{-\beta \Delta E/2}$ where $\Delta E$ is the energy gap. However, the thermistors does not have sharply defined energy bands, and thus 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 \Delta E} is in doubt. Steinhart and Hart simply used empirical curve fitting with following criteria:

1. a single smoothly varying relationship for the entire $T$ range.
2. no plus-minus effect.
3. linear fitting procedures: least squares may be used.

etc.

From B equation $R=R_{0}e^{B({\frac {1}{T}}-{\frac {1}{T_{0}}})}$ we have

${\frac {1}{T}}={\frac {1}{T_{0}}}+{\frac {1}{B}}\ln(R/R_{0})=a_{0}+a_{1}\ln(R/R_{0})$ and expand it to infinite series, but use only some first terms.

Calibrating using known datapoints

Thermistor calibration using three measused points

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

Example. Some known datapoints: R2 = 16200

• boiling water 100 deg gives R2 = 16200
• room temperature 24 deg R2 = 5885
• freezing point of water R2 = 533

Some Semiconductor Physics

Resistivity: $E=\rho J$ where $\rho$ 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 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)} }

The electron density is calculated as

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 n = \int_{-\infty}^\infty g_c(E) f(E) dE }

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 g_c(E)} is the density of states.

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).

Intrinsic conduction (no doped): 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 \ln RT = A + B/T} . It was found that 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} (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 \beta} ) is not constant but dependent on temperature. For intrinsic 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 n=p} giving

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} N_C e^{\beta(E_f - E_c)} &= N_v e^{\beta(E_v - E_f)}\\ E_f &= \frac12 (E_c + E_v) + \frac12 \beta \ln \frac{N_v}{N_c} \end{align} }

and if 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 N_v\approx N_c} , then 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_f} is in the middle of the gap giving

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 n = p = \sqrt{N_c N_v}e^{-\beta \Delta E/2 } }

This will give

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_e = \frac1{B e^{-\beta \Delta E/2}} = Ae^{\beta \Delta E/2} }

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 \ln \rho_e = \ln (\beta \Delta E/2 ) + \ln A }

Target:

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_\textrm{ref} e^{A + B/T + C/T^2 + D/T^3} }

References

• S. Steinhart and S.R. Hart. 1968. "Calibration Curves for Thermistors," Deep-Sea Research 15: 497
• https://www.qtisensing.com/wp-content/uploads/Beta-vs-Steinhart-Hart.pdf
• https://community.element14.com/challenges-projects/design-challenges/experimenting-with-thermistors/b/challenge-blog/posts/blog-4-characterising-thermistors-an-inconvenient-truth-taking-things-to-the-fifth-degree
• https://electronics.stackexchange.com/questions/51865/how-to-get-a-b-and-c-values-for-this-thermistor
• https://www.newport.com/medias/sys_master/images/images/h67/hc1/8797049487390/AN04-Thermistor-Calibration-and-Steinhart-Hart.pdf
• https://www.tdk-electronics.tdk.com/download/531116/19643b7ea798d7c4670141a88cd993f9/%20pdf-general-technical-information.pdf
• https://courses.cit.cornell.edu/ece533/Lectures/handout1.pdf
• http://www.engineeringphysics.weebly.com/uploads/8/2/4/3/8243106/unit_iv_semiconductors.pdf