Update Content - 2025-01-17

content/zettels/temperature_sensors.md

@@ -189,6 +189,68 @@ The huge advantage of RTD compared to PT100 is that the sensitivity is much larg
 {{< 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" >}}
 
 
+## Compute temperature from the measured resistance {#compute-temperature-from-the-measured-resistance}
+
+
+### Pt100 and Pt1000 {#pt100-and-pt1000}
+
+The resistance as a function of temperature is approximated by the Callendar–Van Dusen equation:
+\\[ R(T) = \begin{cases}
+    R\_0 (1 + A \cdot T + B \cdot T^2), & \text{for } T>0^oC \\\\
+    R\_0 (1 + A\cdot T + B \cdot T^2 + C \cdot (T - 100) \cdot T^3), & \text{for } T<0^oC
+\end{cases} \\]
+
+With \\(R\_0\\) the resistance value at 0 degrees (\\(100\\,\Omega\\) for a Pt100 and \\(1000\\,\Omega\\) for a Pt1000).
+
+Values for A, B, C and D are depending on the exact model.
+For a TCR of 3850 ppm/K, the values are:
+
+-   \\(A = 3.9083 \cdot 10^{-3}\ [{}^oC^{-1}]\\)
+-   \\(B = -5.775 \cdot 10^{-7}\ [{}^oC^{-2}]\\)
+-   \\(C = -4.183 \cdot 10^{-12}\ [{}^oC^{-4}]\\)
+
+<!--listend-->
+
+```matlab
+%% Pt100
+R0 = 100; % [Ohm]
+
+A = 3.9083e-3;
+B = -5.775e-7;
+C = -4.183e-12;
+
+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 [$\Omega$]')
+```
+
+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.
+
+```matlab
+%% Compute the temperature as a function of the resistance
+R_meas_1 = 18:100;
+R_meas_2 = 100:390;
+
+
+T_meas = [(-A + sqrt(A^2 - 4*B*(1 - R_meas_2/R0)))/(2*B)];
+
+figure;
+plot(R_meas_2, T_meas)
+xlabel('Resistance [$\Omega$]')
+ylabel('Temperature [${}^oC$]');
+```
+
+
 ## Commercial Temperature Sensors {#commercial-temperature-sensors}