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content/zettels/heat_transfer.md
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@@ -8,52 +8,107 @@ Tags
:
## Electrical Analogy - Lumped Mass Modeling {#electrical-analogy-lumped-mass-modeling}
The difference in temperature \\(\Delta T\\) is driving potential energy flow \\(Q\\) (in watts):
\begin{equation}
\Delta T = R\_{th} \cdot Q
\end{equation}
\\(R\_{th}\\) is the analogy of a "thermal resistance", and is expressed in K/W.
## Conduction (diffusion) {#conduction--diffusion}
The _conduction_ corresponds to the heat transfer \\(P\\) (in watt) through molecular agitation within a material and is specified with:
\\[ P = \frac{\lambda \cdot A \cdot \Delta T}{L} \quad [W] \\]
The _conduction_ corresponds to the heat transfer \\(Q\\) (in watt) through molecular agitation within a material.
\begin{equation}
R\_{th} = \frac{d}{\lambda A}
\end{equation}
with:
- \\(\lambda\\) the thermal conductivity in \\([W/m \cdot K]\\)
- \\(A\\) the surface area in \\([m^2]\\)
- \\(\Delta T\\) the temperature difference in \\([K]\\)
- \\(L\\) the length of the barrier in \\([m]\\)
- \\(d\\) the length of the barrier in \\([m]\\)
## Convection {#convection}
The convection corresponds to the heat transfer through flow of a fluid.
The convection corresponds to the heat transfer \\(Q\\) through flow of a fluid.
It can be either _natural_ or _forced_.
The _forced convection_ \\(P\\) (in watt) can be described with:
\\[ P = h A (T\_0 - T\_f) \quad [W] \\]
\begin{equation}
R\_{th} = \frac{1}{h A}
\end{equation}
with:
- \\(h\\) the convection heat transfer coefficient in \\([W/m^2 \cdot K]\\).
\\(h \approx 10.5 - v + 10\sqrt{v}\\) with \\(v\\) the velocity of the object through the fluid in \\([m/s]\\)
Typically:
- \\(h = 5 - 10\ W/m^2/K\\) for free convection with air
- \\(h = 500 - 5000\ W/m^2/K\\) for forced water cooling in a tube of 5mm diameter
- \\(A\\) the surface area in \\([m^2]\\)
- \\(T\_0\\) the temperature of the object in \\([K]\\)
- \\(T\_f\\) the temperature of the convecting fluid in \\([K]\\)
Note that clean-room air flow should be considered as forced convection.
Note that clean-room air flow should be considered as forced convection, and \\(h \approx 10 W/m^2/K\\).
## Radiation {#radiation}
_Radiation_ corresponds to the heat transfer \\(P\\) (in watt) through the emission of electromagnetic waves from the emitter to its surroundings is:
\\[ P = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) \\]
_Radiation_ corresponds to the heat transfer \\(Q\\) (in watt) through the emission of electromagnetic waves from the emitter to its surroundings.
In the general case, we have:
\\[ Q = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) \\]
with:
- \\(\epsilon\\) the emissivity which corresponds to the ability of a surface to emit energy through radiation relative to a black body surface at equal temperature.
It is between 0 (no emissivity) and 1 (maximum emissivity)
- \\(\sigma\\) the Stefan-Boltzmann constant: \\(\sigma = 5.67 \cdot 10^{-8} \\, \frac{W}{m^2 K^4}\\)
- \\(A\\) the surface in \\([m^2]\\)
- \\(T\_r\\) the temperature of the emitter in \\([K]\\)
- \\(T\_s\\) the temperature of the surrounding in \\([K]\\)
In order to use the lumped mass approximation, the equations can be linearized to obtain:
\begin{equation}
R\_{th} = \frac{1}{h\_{rad} A}
\end{equation}
with:
- \\(h\_{rad}\\) the effective heat transfer coefficient for radiation in \\(W/m^2 \cdot K\\)
- \\(A\\) the surface in \\([m^2]\\)
### Practical Cases {#practical-cases}
Two parallel plates:
\begin{equation}
h\_{rad} = \frac{\sigma}{1/\epsilon\_1 + 1/\epsilon\_2 - 1} (T\_1^2 + T\_2^2)(T\_1 + T\_2)
\end{equation}
Two concentric cylinders:
\begin{equation}
h\_{rad} = \frac{\sigma}{1/\epsilon\_1 + r\_1/r\_2 (1/\epsilon\_2 - 1)} (T\_1^2 + T\_2^2)(T\_1 + T\_2)
\end{equation}
A small object enclosed in a large volume:
\begin{equation}
h\_{rad} = \epsilon\_1 \sigma (T\_1^2 + T\_2^2)(T\_1 + T\_2)
\end{equation}
### Emissivity {#emissivity}
The emissivity of materials highly depend on the surface finish (the more polished, the lower the emissivity).
Some examples are given in <tab:emissivity_examples>.
Gold coating gives also a very low emissivity and is typically used in cryogenic applications.
<a id="table--tab:emissivity-examples"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:emissivity-examples">Table 1</a>:</span>
@@ -114,6 +169,35 @@ The required heat is then:
| Water at 25 degrees | 4182 |
## Heat Transport (i.e. Water cooling) {#heat-transport--i-dot-e-dot-water-cooling}
<a id="figure--fig:heat-transfer-fluid"></a>
{{< figure src="/ox-hugo/heat_transfer_fluid.png" caption="<span class=\"figure-number\">Figure 1: </span>Heat transfered to the fluid" >}}
\begin{equation}
Q\_{in} = h \cdot A \cdot (T\_{wall} - T\_{mean})
\end{equation}
<a id="figure--fig:heat-transport"></a>
{{< figure src="/ox-hugo/heat_transport.png" caption="<span class=\"figure-number\">Figure 2: </span>Heat Transport in the fluid" >}}
\begin{equation}
Q\_{out} = \phi \rho c\_p (T\_{mean,in} - T\_{mean,out})
\end{equation}
with:
- \\(Q\_{out}\\) the transported heat in W
- \\(\phi\\) the flow in \\(m^3/s\\)
- \\(\rho\\) the fluid density in \\(kg/m^3\\)
- \\(c\_p\\) the specific heat capacity of the fluid in \\(J/(kg \cdot K)\\)
- \\(T\_{mean}\\) the mean incoming and outgoing fluid temperature
Because of energy balance, we have in the stationary condition: \\(Q\_{in} = Q\_{out}\\)
## Heat flow {#heat-flow}
The heat flow \\(P\\) (in watt) is the derivative of the heat:

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content/zettels/temperature_sensors.md
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@@ -10,29 +10,13 @@ Tags
## Temperature sensors types {#temperature-sensors-types}
There are three main types of temperature sensors:
There are mainly three main types of temperature sensors:
- [Thermocouples](#org-target--sec-temperature-sensor-thermocouple) with are based on the Seebeck effect
- [RTD](#org-target--sec-temperature-sensor-rtd) (Resistance Temperature Detectors): made of pure metals (Pt, Ni or Cu)
They are all PTC (Positive Temperature Coefficient): PT100, PT1000, Ni100, Ni1000, ...
- [Thermistor](#org-target--sec-temperature-sensor-thermistor): made of metal oxide mixtures (semiconductor materials).
It can have a NTC (Negative Temperature Coefficient) or a PTC (Positive Temperature Coefficient).
- [Thermocouple](#org-target--sec-temperature-sensor-thermocouple) (Seebeck effect):
- Type K (-180 to 1200 degC)
- Type T (-250 to 350 degC)
### RTD {#rtd}
<span class="org-target" id="org-target--sec-temperature-sensor-rtd"></span>
Sensitivity of PT100 is typically around 0.004 Ohm/Ohm/deg.
### Thermistor {#thermistor}
<span class="org-target" id="org-target--sec-temperature-sensor-thermistor"></span>
Sensitivity of NTC is typically around -0.05 Ohm/Ohm/deg.
They typically have a NTC (Negative Temperature Coefficient).
### Thermocouple {#thermocouple}
@@ -40,17 +24,134 @@ Sensitivity of NTC is typically around -0.05 Ohm/Ohm/deg.
<span class="org-target" id="org-target--sec-temperature-sensor-thermocouple"></span>
#### Working Principle {#working-principle}
Based on Seebeck effect: in a conductor, a temperature difference \\(\Delta T\\) creates an electric field \\(V\\) defined by the Seebeck coefficient \\(\epsilon\\).
Consider a material with a Seebeck coefficient \\(\epsilon\_A\\), the voltage difference at its send is:
\begin{equation}
V\_A = \int\_{T\_1}^{T\_2} \epsilon\_A dT = \epsilon\_A (T\_2 - T\_1)
\end{equation}
<a id="figure--fig:temperature-seebeck-material-A"></a>
{{< figure src="/ox-hugo/temperature_seebeck_material_A.png" caption="<span class=\"figure-number\">Figure 1: </span>Seebeck effect in a material" >}}
Now if two materials are "chained", the overall voltage will be:
\begin{equation}
V = (\epsilon\_{A} - \epsilon\_{B})(T\_{2} - T\_{1})
\end{equation}
<a id="figure--fig:temperature-thermocouple-two-materiasl"></a>
{{< figure src="/ox-hugo/temperature_thermocouple_two_materiasl.png" caption="<span class=\"figure-number\">Figure 2: </span>Combination of two materials" >}}
Several materials can be combined:
<a id="figure--fig:temperature-termocouples-sensitivities"></a>
{{< figure src="/ox-hugo/temperature_termocouples_sensitivities.png" caption="<span class=\"figure-number\">Figure 3: </span>Some types of thermocouples and associated sensitivities" >}}
#### Measurement of the temperature {#measurement-of-the-temperature}
The output voltage of a thermocouple is given by \\(\Delta T\\) (between \\(T\_1\\), the temperature at the measured voltage side, and \\(T\_2\\) at the thermocouple side).
Therefore, typically \\(T\_1\\) is measured with an RTD (typically what is done inside the Agilent 34970A) and \\(T\_2\\) is therefore estimated from \\(T\_1\\) and \\(\Delta T\\).
<a id="figure--fig:temperature-thermocouple-meas"></a>
{{< figure src="/ox-hugo/temperature_thermocouple_meas.png" caption="<span class=\"figure-number\">Figure 4: </span>Thermocouple measured voltage" >}}
#### Choice of Thermocouple {#choice-of-thermocouple}
- J - Iron / Constantan
- \\(55\ \mu V / K\\)
- Easy to solder
- Can act as galvanic element
- K - Chromel / Alumel
- \\(40\ \mu V / K\\), almost linear
- Difficult to solver, welding is better
- Low thermal conductivity of wires
- T - Copper / Constantan
- \\(40\ \mu V / K\\)
- Easy to solder
- High thermal conductivity is source of errors
#### Summary {#summary}
Advantages:
- Simple to use
- Standard acquisition systems
- Small (thin wires down to 0.08mm)
- Fast response
- Suitable for high and low temperatures
- Can be used in vacuum systems
- No self heating
Disadvantages:
- Relative expensive
- Uncertainty about 0.1K, not feasible to reach mK
### RTD {#rtd}
<span class="org-target" id="org-target--sec-temperature-sensor-rtd"></span>
Sensitivity of PT100 is typically around .
Advantages:
- Very high stability (better than 1mK per year)
- Very high linearity (0.1% over -40 to 125 degrees)
- This makes them very useful as calibration reference sensor, which is linked to the international standard
Disadvantages:
- Expensive
- Low sensitivity (typically 0.004 Ohm/Ohm/deg)
- The measurement is sensitivity to lead wire resistance, but four-wire technique may be used
- Self heating due to electrical dissipation.
Typically, for a Pt100, \\(P = 0.1 mW\\) (the source voltage is typically 0.1 V)
This corresponds to approximately 0.1 degree of self heating in "still" air
### Thermistor {#thermistor}
<span class="org-target" id="org-target--sec-temperature-sensor-thermistor"></span>
Sensitivity of NTC is .
Advantages:
- Highest sensitivity (typically around -0.05 Ohm/Ohm/deg)
- Because the resistance is typically high (100k Ohm), no Four wire configuration is necessary, and long wires may be used
- Lower heat dissipation than Pt100
- Very high stability, especially for glass encapsulated
- Very small, and available in all kinds of shapes
Disadvantages:
- Non-linear, so compensation is necessary (but not really an issue with software compensation)
- Self-heating if mK accuracy is wanted
- Cover is necessary for use in vacuum
### Comparison of sensor types {#comparison-of-sensor-types}
<https://www.ni.com/fr/shop/data-acquisition/sensor-fundamentals/measuring-temperature-with-thermocouples-rtds-and-thermistors.html?srsltid=AfmBOoqrR2VCWVvkCTjzTIYHlyiKVpt6Ket1xfhU1yFPTtXsGKg0RILE>
| | RTD | Thermistor | Thermocouple |
|---------------|------|------------|--------------|
| Accuracy | Good | Non-Linear | |
| Stability | | | |
| Sensitivity | Bad | Good | |
| Response time | | | |
| Self heating | | | |
| | RTD | Thermistor | Thermocouple | |
|---------------|-------------------------|------------|--------------|---|
| Accuracy | Good | Non-Linear | | A |
| Linearity | 0.1% over -40..125 degC | | | |
| Stability | better than 1mK/year | | | |
| Sensitivity | 0.4%/K | 5%/K | 50uV/K | |
| Response time | | | | |
| Self heating | | | None | |
## Accuracy of Temperature measurement {#accuracy-of-temperature-measurement}
@@ -73,7 +174,7 @@ For a Pt100 at \\(0^oC\\), this corresponds to an accuracy of \\(< \pm 0.04\\,K\
#### 2, 3 and 4 wires sensors {#2-3-and-4-wires-sensors}
The measured resistance is the sum of the resistance of the sensitive element and the resistance of the wires.
This corresponds to the 2-wire measurement ([Figure 1](#figure--fig:temperature-sensor-rtd-2-wires)).
This corresponds to the 2-wire measurement ([Figure 5](#figure--fig:temperature-sensor-rtd-2-wires)).
The errors associated with this effect are large when the resistance of the sensitive element is small and then the resistance of all cables and connectors are large.
For instance, the effect of contact/wire resistance less important for the PT1000 than for the PT100.
@@ -81,18 +182,27 @@ The use of 2 wire PT1000 is possible (whereas for PT100, 4 wire is more accurate
<a id="figure--fig:temperature-sensor-rtd-2-wires"></a>
{{< figure src="/ox-hugo/temperature_sensor_rtd_2_wires.png" caption="<span class=\"figure-number\">Figure 1: </span>2-wire measurement" >}}
{{< figure src="/ox-hugo/temperature_sensor_rtd_2_wires.png" caption="<span class=\"figure-number\">Figure 5: </span>2-wire measurement" >}}
The effect of the resistance of the wires (cables, connectors, etc..) can be mitigated by using the 4-wire configuration ([Figure 2](#figure--fig:temperature-sensor-rtd-4-wires)).
The effect of the resistance of the wires (cables, connectors, etc..) can be mitigated by using the 4-wire configuration ([Figure 6](#figure--fig:temperature-sensor-rtd-4-wires)).
<a id="figure--fig:temperature-sensor-rtd-4-wires"></a>
{{< figure src="/ox-hugo/temperature_sensor_rtd_4_wires.png" caption="<span class=\"figure-number\">Figure 2: </span>4-wire measurement" >}}
{{< figure src="/ox-hugo/temperature_sensor_rtd_4_wires.png" caption="<span class=\"figure-number\">Figure 6: </span>4-wire measurement" >}}
### Temperature {#temperature}
#### Effect of conductivity through the wires {#effect-of-conductivity-through-the-wires}
It is better to use thin wires, of the fix the wires to the part that is to be measured.
<a id="figure--fig:temperature-effect-wires"></a>
{{< figure src="/ox-hugo/temperature_effect_wires.png" caption="<span class=\"figure-number\">Figure 7: </span>Measured effect of wires. When in \"air\", it conducts the heat from the air to the sensor which can lead to measurement errors." >}}
#### Thermal contact and response time {#thermal-contact-and-response-time}
The measured temperature is the temperature of the sensitive element.
@@ -100,6 +210,10 @@ It may not be equal to the temperature of the element on which the sensor is fix
It depends on the thermal contact and the response time in play.
The sensor contact may be improved by using "soft" (i.e. plastically deformable) metals at the contact interface such as indium.
However, it seems that having too much pressure in the sensor may induce stress in the NTC that can induce measurement errors.
#### Self heating effect {#self-heating-effect}
@@ -113,12 +227,14 @@ Typically, a constant voltage is applied, such that the generated current is low
<div class="exampl">
The applied voltage is typically in the order of 1V.
For a Pt100 (\\(R \approx 100\\,\Omega\\)), this would lead a heat generation of \\(P \approx 10\\,mW\\).
For a NTC with \\(R\approx 10\\,k\Omega\\), the heat generation will me much lower \\(P\approx 100\\,\mu W\\).
The applied voltage is typically in the order of 0.1V to 1V.
For a Pt100 (\\(R \approx 100\\,\Omega\\)), this would lead a heat generation of \\(P \approx 1 \text{ to } 10\\,mW\\).
For a NTC with \\(R\approx 10\\,k\Omega\\), the heat generation will me much lower \\(P\approx 10 \text{ to } 100\\,\mu W\\).
</div>
In order to lower the self heating effect, _intermitted_ currents may be used as is the case with the Agilent 34970A.
### Converting Resistance to Temperature {#converting-resistance-to-temperature}
@@ -150,26 +266,26 @@ Manufacturers usually provides a lookup table that links the resistance and the
### PT100 {#pt100}
A PT100 resistance is quite linear with respect to the temperature as shown in [Figure 3](#figure--fig:temperature-sensor-pt100-resistance).
A PT100 resistance is quite linear with respect to the temperature as shown in [Figure 8](#figure--fig:temperature-sensor-pt100-resistance).
<a id="figure--fig:temperature-sensor-pt100-resistance"></a>
{{< figure src="/ox-hugo/temperature_sensor_pt100_resistance.png" caption="<span class=\"figure-number\">Figure 3: </span>Resistance of a PT100 as a function of the temperature" >}}
{{< figure src="/ox-hugo/temperature_sensor_pt100_resistance.png" caption="<span class=\"figure-number\">Figure 8: </span>Resistance of a PT100 as a function of the temperature" >}}
The coefficient of resistance \\(\alpha\\) is defined as the ratio of the rate of change of resistance with temperature to the resistance of the thermistor at a specified temperature:
\\[ \alpha(T) = \frac{1}{R(T)}\frac{dR(T)}{dT} \\]
For a PT100, it is displayed in [Figure 4](#figure--fig:temperature-sensor-pt100-sensitivity).
For a PT100, it is displayed in [Figure 9](#figure--fig:temperature-sensor-pt100-sensitivity).
At \\(0^oC\\), \\(\alpha(0^oC) \approx 0.004\\,\Omega/\Omega/{}^oC\\).
<a id="figure--fig:temperature-sensor-pt100-sensitivity"></a>
{{< figure src="/ox-hugo/temperature_sensor_pt100_sensitivity.png" caption="<span class=\"figure-number\">Figure 4: </span>Sensitivity of a PT100 as a function of the temperature" >}}
{{< figure src="/ox-hugo/temperature_sensor_pt100_sensitivity.png" caption="<span class=\"figure-number\">Figure 9: </span>Sensitivity of a PT100 as a function of the temperature" >}}
### NTC {#ntc}
A NTC is much more non-linear than a PT100 as shown in [Figure 5](#figure--fig:temperature-sensor-rtd-resistance).
A NTC is much more non-linear than a PT100 as shown in [Figure 10](#figure--fig:temperature-sensor-rtd-resistance).
The NTC used here is "Type F" from Amphenol Thermometrics.
@@ -180,13 +296,13 @@ R_rtd = 1e4*[68.60 48.16 34.23 24.62 17.91 13.17 9.782 7.339 5.558 4.247 3.274 2
<a id="figure--fig:temperature-sensor-rtd-resistance"></a>
{{< figure src="/ox-hugo/temperature_sensor_rtd_resistance.png" caption="<span class=\"figure-number\">Figure 5: </span>Resistance of a RTD as a function of the temperature" >}}
{{< figure src="/ox-hugo/temperature_sensor_rtd_resistance.png" caption="<span class=\"figure-number\">Figure 10: </span>Resistance of a RTD as a function of the temperature" >}}
The huge advantage of RTD compared to PT100 is that the sensitivity is much larger than Pt100 as shown in [Figure 6](#figure--fig:temperature-sensor-rtd-sensitivity).
The huge advantage of RTD compared to PT100 is that the sensitivity is much larger than Pt100 as shown in [Figure 11](#figure--fig:temperature-sensor-rtd-sensitivity).
<a id="figure--fig:temperature-sensor-rtd-sensitivity"></a>
{{< figure src="/ox-hugo/temperature_sensor_rtd_sensitivity.png" caption="<span class=\"figure-number\">Figure 6: </span>Sensitivity of a RTD as a function of the temperature" >}}
{{< figure src="/ox-hugo/temperature_sensor_rtd_sensitivity.png" caption="<span class=\"figure-number\">Figure 11: </span>Sensitivity of a RTD as a function of the temperature" >}}
## Compute temperature from the measured resistance {#compute-temperature-from-the-measured-resistance}
@@ -233,7 +349,7 @@ R = [R0*(1 + A*T1 + B*T1.^2 + C*(T1-100).*T1.^3), R0*(1 + A*T2 + B*T2.^2)]; % [O
<a id="figure--fig:temperature-sensor-pt100-curve"></a>
{{< figure src="/ox-hugo/temperature_sensor_pt100_curve.png" caption="<span class=\"figure-number\">Figure 7: </span>Resistance as a function of the temperature for a Pt100" >}}
{{< figure src="/ox-hugo/temperature_sensor_pt100_curve.png" caption="<span class=\"figure-number\">Figure 12: </span>Resistance as a function of the temperature for a Pt100" >}}
For temperatures above 0 degrees, the temperature \\(T\\) can be easily computed from the measured resistance \\(R\\) using:
\\[ T = \frac{-A + \sqrt{A^2 - 4 B ( 1 - R/R\_0 )}}{2 B} \\]
@@ -245,7 +361,7 @@ Let's compare the temperature given by a Loopup table and the temperature given
- linear interpolation with one point every degree
- cubic interpolation with one point every 10 degrees
The error is less than 0.1mK over the full range, validating the use of a lookup table to convert the resistance to temperature ([Figure 9](#figure--fig:temperature-sensor-lut-errors)).
The error is less than 0.1mK over the full range, validating the use of a lookup table to convert the resistance to temperature ([Figure 14](#figure--fig:temperature-sensor-lut-errors)).
### NTC thermistor {#ntc-thermistor}
@@ -288,7 +404,7 @@ T = -273.15+[T1,T2,T3,T4]; % [degC]
<a id="figure--fig:temperature-sensor-ntc-curve"></a>
{{< figure src="/ox-hugo/temperature_sensor_ntc_curve.png" caption="<span class=\"figure-number\">Figure 8: </span>Resistance as a function of the temperature for a given NTC" >}}
{{< figure src="/ox-hugo/temperature_sensor_ntc_curve.png" caption="<span class=\"figure-number\">Figure 13: </span>Resistance as a function of the temperature for a given NTC" >}}
To calculate the actual thermistor temperature as a function of the measured thermistor resistance, use the following equation:
\\[ T = \frac{1}{a + b \ln(R\_t/R\_{25}) + c (Ln Rt/R25)^2 + d (Ln Rt/R25)^3) \\]
@@ -309,7 +425,7 @@ To calculate the actual thermistor temperature as a function of the measured the
### Approximation of formulas using lookup tables {#approximation-of-formulas-using-lookup-tables}
First, let's compare the analytical formula with a LUT for a Pt100 ([Figure 9](#figure--fig:temperature-sensor-lut-errors)).
First, let's compare the analytical formula with a LUT for a Pt100 ([Figure 14](#figure--fig:temperature-sensor-lut-errors)).
The error (accuracy) is bellow 0.1mK for relatively small LUT.
```matlab
@@ -349,10 +465,10 @@ T_meas_makima = interp1(R_lut_makima,T_lut_makima,R_true,interp_method); % inter
<a id="figure--fig:temperature-sensor-lut-errors"></a>
{{< figure src="/ox-hugo/temperature_sensor_lut_errors.png" caption="<span class=\"figure-number\">Figure 9: </span>Interpolation errors in two cases when using a LUT for a Pt100" >}}
{{< figure src="/ox-hugo/temperature_sensor_lut_errors.png" caption="<span class=\"figure-number\">Figure 14: </span>Interpolation errors in two cases when using a LUT for a Pt100" >}}
NTC thermistors are more non-linear and therefore require finer LUT to have low accuracy errors.
In order to have less than 0.1mK of accuracy, a LUT with linear interpolation requires approximately one point every 0.1 degree ([Figure 10](#figure--fig:temperature-sensor-lut-errors-ntc)).
In order to have less than 0.1mK of accuracy, a LUT with linear interpolation requires approximately one point every 0.1 degree ([Figure 15](#figure--fig:temperature-sensor-lut-errors-ntc)).
```matlab
%% "Perfect" temperature and resistance of NTC (DC95F202VN)
@@ -406,7 +522,7 @@ T_meas_makima = interp1(R_lut_makima,T_lut_makima,R_true,interp_method); % inter
<a id="figure--fig:temperature-sensor-lut-errors-ntc"></a>
{{< figure src="/ox-hugo/temperature_sensor_lut_errors_ntc.png" caption="<span class=\"figure-number\">Figure 10: </span>Interpolation errors in two cases when using a LUT for a NTC" >}}
{{< figure src="/ox-hugo/temperature_sensor_lut_errors_ntc.png" caption="<span class=\"figure-number\">Figure 15: </span>Interpolation errors in two cases when using a LUT for a NTC" >}}
## Commercial Temperature Sensors {#commercial-temperature-sensors}
@@ -438,10 +554,28 @@ From (<a href="#citeproc_bib_item_2">Neto et al. 2022</a>)
> Furthermore, the thin platinum wire of the 10 kΩ RTDs presented bad solderability and its assembly process was too laborious, resulting in unreliable mechanical bonds and a failure rate beyond acceptable for a robust beamline instrumentation.
> The alternative was to use **2 kΩ IST RTDs (P2K0.232.3FW.B.007)** with custom-made flat gold-platted terminals, resulting in a full range sensor with better solderability and temperature **resolution below 0.4 mK** over the entire measurable range
**NTC, Amphenol DC95F**, measured with an Agilent 34970A at 10K:
- leads to 0.2mK resolution (22 bits)
- High interchangeability: offset of 0.01K and sensitivity of 3mK/K
<a id="figure--fig:temperature-ntc-dc95f-results"></a>
{{< figure src="/ox-hugo/temperature_ntc_dc95f_results.png" caption="<span class=\"figure-number\">Figure 16: </span>6 (un-calibrated) DC95F sensors fixed to the same mass with homogeneous temperature" >}}
**NTC, Betatherm 10K3A1**, measured at 10K:
- Resolution of 0.2mK
- Low noise and high repeatability
<a id="figure--fig:temperature-betatherm-10K3A1-results"></a>
{{< figure src="/ox-hugo/temperature_betatherm_10K3A1_results.png" caption="<span class=\"figure-number\">Figure 17: </span>Measured temperature of two BetaTherm 10K3A1 compared to a reference sensor, at 10K" >}}
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Ebrahimi-Darkhaneh, Hadi. 2019. “Measurement Error Caused by Self-Heating in Ntc and Ptc Thermistors.” <i>Tex. Instrum. Analog. Des. J. Q</i> 3: 001–7.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Neto, Joao Brito, Renan Geraldes, Francesco Lena, Marcelo Moraes, Antonio Piccino Neto, Marlon Saveri Silva, and Lucas Volpe. 2022. “Temperature Control for Beamline Precision Systems of Sirius/Lnls.” <i>Proceedings of the 18th International Conference on Accelerator and Large Experimental Physics Control Systems</i> ICALEPCS2021 (nil): China. doi:<a href="https://doi.org/10.18429/JACOW-ICALEPCS2021-WEPV001">10.18429/JACOW-ICALEPCS2021-WEPV001</a>.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Neto, Joao Brito, Renan Geraldes, Francesco Lena, Marcelo Moraes, Antonio Piccino Neto, Marlon Saveri Silva, and Lucas Volpe. 2022. “Temperature Control for Beamline Precision Systems of Sirius/Lnls.” <i>Proceedings of the 18th International Conference on Accelerator and Large Experimental Physics Control Systems</i> ICALEPCS2021: China. doi:<a href="https://doi.org/10.18429/JACOW-ICALEPCS2021-WEPV001">10.18429/JACOW-ICALEPCS2021-WEPV001</a>.</div>
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