Update Content - 2025-01-17

content/zettels/temperature_sensors.md

@@ -220,41 +220,131 @@ Values for A, B, C and D are depending on the exact model (summarized in [Table
 %% Pt100 (3850 ppm/K)
 R0 = 100; % [Ohm]
 
-A = 3.9083e-3;
+A = 3.9083e-3; % [degC^-1]
-B = -5.775e-7;
+B = -5.775e-7; % [degC^-2]
-C = -4.183e-12;
+C = -4.183e-12; % [degC^-4]
 
 T1 = -200:0; % [degC]
 T2 = 0:850; % [degC]
 T = [T1,T2]; % [degC]
 
 R = [R0*(1 + A*T1 + B*T1.^2 + C*(T1-100).*T1.^3), R0*(1 + A*T2 + B*T2.^2)]; % [Ohm]
-
-figure;
-plot(T, R)
-xlabel('Temperature [${}^oC$]');
-ylabel('Resistance [$\Omega$]')
 ```
 
+<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" >}}
+
 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} \\]
 
 For temperatures below 0 degrees, the equation is harder to solve analytically, and a lookup table is more appropriate.
 
+Let's compare the temperature given by a Loopup table and the temperature given by the analytical formula in two cases:
+
+-   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 8](#figure--fig:temperature-sensor-lut-errors)).
+
 ```matlab
-%% Compute the temperature as a function of the resistance
+%% "Perfect" temperature and resistance
-R_meas_1 = 18:100;
+R0 = 100; % [Ohm]
-R_meas_2 = 100:390;
+A = 3.9083e-3; % [degC^-1]
+B = -5.775e-7; % [degC^-2]
+C = -4.183e-12; % [degC^-4]
 
+T1 = -200:0.1:0; % [degC]
+T2 = 0.1:0.1:850; % [degC]
+T_true = [T1,T2]; % [degC]
+R_true = [R0*(1 + A*T1 + B*T1.^2 + C*(T1-100).*T1.^3), R0*(1 + A*T2 + B*T2.^2)]; % [Ohm]
 
-T_meas = [(-A + sqrt(A^2 - 4*B*(1 - R_meas_2/R0)))/(2*B)];
+%% Lookup table for Pt100 (3850 ppm/K) - Linear
+dT = 1;
+interp_method = 'linear';
 
-figure;
+T1 = -200:dT:0; % [degC]
-plot(R_meas_2, T_meas)
+T2 = dT:dT:850; % [degC]
-xlabel('Resistance [$\Omega$]')
+T_lut_linear = [T1,T2]; % [degC]
-ylabel('Temperature [${}^oC$]');
+R_lut_linear = [R0*(1 + A*T1 + B*T1.^2 + C*(T1-100).*T1.^3), R0*(1 + A*T2 + B*T2.^2)]; % [Ohm]
+
+T_meas_linear = interp1(R_lut_linear,T_lut_linear,R_true,interp_method); % interpolate the resistance using the LUT to find the corresponding temperature
+
+%% Lookup table for Pt100 (3850 ppm/K) - Makima
+dT = 10;
+interp_method = 'makima';
+
+T1 = -200:dT:0; % [degC]
+T2 = dT:dT:850; % [degC]
+T_lut_makima = [T1,T2]; % [degC]
+R_lut_makima = [R0*(1 + A*T1 + B*T1.^2 + C*(T1-100).*T1.^3), R0*(1 + A*T2 + B*T2.^2)]; % [Ohm]
+
+T_meas_makima = interp1(R_lut_makima,T_lut_makima,R_true,interp_method); % interpolate the resistance using the LUT to find the corresponding temperature
 ```
 
+<a id="figure--fig:temperature-sensor-lut-errors"></a>
+
+{{< figure src="/ox-hugo/temperature_sensor_lut_errors.png" caption="<span class=\"figure-number\">Figure 8: </span>Interpolation errors in two cases when using a LUT for a Pt100" >}}
+
+
+### NTC thermistor {#ntc-thermistor}
+
+The resistance of the NTC thermistor as a function of the temperature can be well approximated with the following equation:
+\\[ R\_t = R\_{25} \cdot e^{A + B/T + C/T^2 + D/T^3 \\]
+where \\(T\\) is the temperature in kelvins, \\(R\_{25}\\) the nominal resistance at \\(25^oC\\), \\(A\\), \\(B\\), \\(C\\) and \\(D\\) are coefficients which are specific for a given thermistor.
+
+Typically, coefficients A, B, C and D are varying with temperature as shown in [Table 2](#table--tab:temperature-sensor-ntc-coefs).
+
+<a id="table--tab:temperature-sensor-ntc-coefs"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:temperature-sensor-ntc-coefs">Table 2</a>:</span>
+  Example of A, B, C and D coeficients for an NTC thermistor (DC95F202VN)
+</div>
+
+|            | A              | B             | C              | D              |
+|------------|----------------|---------------|----------------|----------------|
+| -50 to 0   | -1.4122478E+01 | 4.4136033E+03 | -2.9034189E+04 | -9.3875035E+06 |
+| 0 to 50    | -1.4141963E+01 | 4.4307830E+03 | -3.4078983E+04 | -8.8941929E+06 |
+| 50 to 100  | -1.4202172E+01 | 4.4975256E+03 | -5.8421357E+04 | -5.9658796E+06 |
+| 100 to 150 | -1.6154078E+01 | 6.8483992E+03 | -1.0004049E+06 | 1.1961431E+08  |
+
+```matlab
+%% Compute the resistance as a function of the temperature for a given NTC (DC95F202VN)
+R0 = 2e3; % Resistance at 25deg
+
+T1 = 273.15+[-50:0]; % [degK]
+T2 = 273.15+[1:50]; % [degK]
+T3 = 273.15+[51:100]; % [degK]
+T4 = 273.15+[101:150]; % [degK]
+
+R = R0*exp([[-1.4122478E+01 + 4.4136033E+03./T1 - 2.9034189E+04./T1.^2 - 9.3875035E+06./T1.^3]';
+            [-1.4141963E+01 + 4.4307830E+03./T2 - 3.4078983E+04./T2.^2 - 8.8941929E+06./T2.^3]';
+            [-1.4202172E+01 + 4.4975256E+03./T3 - 5.8421357E+04./T3.^2 - 5.9658796E+06./T3.^3]';
+            [-1.6154078E+01 + 6.8483992E+03./T4 - 1.0004049E+06./T4.^2 + 1.1961431E+08./T4.^3]'])'; % [Ohm]
+
+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 9: </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) \\]
+
+<a id="table--tab:temperature-sensor-ntc-equation"></a>
+<div class="table-caption">
+  <span class="table-number"><a href="#table--tab:temperature-sensor-ntc-equation">Table 3</a>:</span>
+  Coefficients used to compute the temperature as a function of the resistance
+</div>
+
+| Rt/R25 range       | a             | b             | c              | d              |
+|--------------------|---------------|---------------|----------------|----------------|
+| 68.600 to 3.274    | 3.3538646E-03 | 2.5654090E-04 | 1.9243889E-06  | 1.0969244E-07  |
+| 3.274 to 0.36036   | 3.3540154E-03 | 2.5627725E-04 | 2.0829210E-06  | 7.3003206E-08  |
+| 0.36036 to 0.06831 | 3.3539264E-03 | 2.5609446E-04 | 1.9621987E-06  | 4.6045930E-08  |
+| 0.06831 to 0.01872 | 3.3368620E-03 | 2.4057263E-04 | -2.6687093E-06 | -4.0719355E-07 |
+
 
 ## Commercial Temperature Sensors {#commercial-temperature-sensors}