NTC calibration and multiple temperature sensors: Difference between revisions

Latest revision as of 09:14, 28 March 2024

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 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.

B equation

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

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_0 e^{B(\frac1T - \frac1{T_0})} }

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(R) + C (\ln (R))^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.

Steinhart-Hart Equation was found in 1968.

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^{-1} = F(T) e^{- \beta \Delta E/2} } 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 \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:

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

etc.

From B equation 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_0 e^{B(\frac1T - \frac1{T_0})} } we have

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 \frac1T = \frac1{T_0} + \frac1B \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

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

LM35DZ

GY-91

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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n=\int _{-\infty }^{\infty }g_{c}(E)f(E)dE}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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): $\ln RT=A+B/T$ . It was found that $B$ ( $\beta$ ) is not constant but dependent on temperature. For intrinsic $n=p$ giving

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}N_{C}e^{\beta (E_{f}-E_{c})}&=N_{v}e^{\beta (E_{v}-E_{f})}\\E_{f}&={\frac {1}{2}}(E_{c}+E_{v})+{\frac {1}{2}}\beta \ln {\frac {N_{v}}{N_{c}}}\end{aligned}}}

and if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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

@@ Line 33: / Line 33: @@
-=== Calibrating: Steinhart-Hart Equation ===
+== B equation ==
+Steinhart--Hart equation can be derived from extending the B parameter equation to an infinite series.
+<math>
+R = R_0 e^{B(\frac1T - \frac1{T_0})}
+</math>
+== Steinhart-Hart Equation ==
 A nonlinear Steinhart-Hart equation is widely used
@@ Line 46: / Line 55: @@
 <math>
-\rho^{-1} = F(T) e^{- \Delta E/2kT}
+\rho^{-1} = F(T) e^{- \beta \Delta E/2}
 </math>
 where <math>\Delta E</math> is the energy gap. However, the thermistors does not have sharply defined  energy bands, and thus <math>\Delta E</math> is in doubt. Steinhart and Hart simply used empirical curve fitting with following criteria:
@@ Line 53: / Line 62: @@
 # linear fitting procedures: least squares may be used.
 etc.
+From B equation
+<math>
+R = R_0 e^{B(\frac1T - \frac1{T_0})}
+</math>
+we have
+<math>
+\frac1T = \frac1{T_0} + \frac1B \ln(R/R_0) = a_0 + a_1 \ln (R/R_0)
+</math>
+and expand it to infinite series, but use only some first terms.
 === Calibrating using known datapoints ===
+[[File:Thermistor calibrate desmos.png|thumb|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.
-Some known datapoints:
+Example. Some known datapoints: R2 = 16200
-* boiling water 100 deg
+* boiling water 100 deg gives  R2 = 16200
-* room temperature
+* room temperature 24 deg R2 = 5885
-* freezing point of water
+* freezing point of water R2 = 533
-=== LM35DZ ==
+== LM35DZ ==
 == GY-91 ==
@@ Line 108: / Line 131: @@
 f_{M-B}(\epsilon)=e^{-\beta(\epsilon-\mu)}
 </math>
+The electron density is calculated as
+<math>
+n = \int_{-\infty}^\infty g_c(E) f(E) dE
+</math>
+where <math>g_c(E)</math> 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).
@@ Line 113: / Line 146: @@
-Intrinsic conduction: <math>\ln RT = A + B/T</math>. It was found that <math>B</math> (<math>\beta</math>) is not constant but dependent on temperature.
+Intrinsic conduction (no doped): <math>\ln RT = A + B/T</math>. It was found that <math>B</math> (<math>\beta</math>) is not constant but dependent on temperature. For intrinsic <math>n=p</math> giving
+<math>
+\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}
+</math>
+and if <math>N_v\approx N_c</math>, then <math>E_f</math> is in the middle of the gap giving
+<math>
+n = p = \sqrt{N_c N_v}e^{-\beta \Delta E/2 }
+</math>
+This will give
+<math>
+\rho_e = \frac1{B e^{-\beta \Delta E/2}} = Ae^{\beta \Delta E/2}
+</math>
+which gives
+<math>
+\ln \rho_e = \ln (\beta \Delta E/2 ) + \ln A
+</math>
@@ Line 121: / Line 180: @@
 R = R_\textrm{ref} e^{A + B/T + C/T^2 + D/T^3}
 </math>
 == References ==
@@ Line 129: / Line 187: @@
 * 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

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Latest revision as of 09:14, 28 March 2024

Contents

Introduction

NTC

B equation

Steinhart-Hart Equation

Calibrating using known datapoints

LM35DZ

GY-91

Some Semiconductor Physics

Target:

References

Navigation

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Anonymous

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NTC calibration and multiple temperature sensors: Difference between revisions

Latest revision as of 09:14, 28 March 2024

Introduction

NTC

B equation

Steinhart-Hart Equation

Calibrating using known datapoints

LM35DZ

GY-91

Some Semiconductor Physics

Target:

References

Navigation

Wiki tools

Page tools

Categories