diff --git a/content/zettels/temperature_sensors.md b/content/zettels/temperature_sensors.md index 8c6ff9a..9268103 100644 --- a/content/zettels/temperature_sensors.md +++ b/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; -B = -5.775e-7; -C = -4.183e-12; +A = 3.9083e-3; % [degC^-1] +B = -5.775e-7; % [degC^-2] +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 [$\Omega$]') ``` + + +{{< figure src="/ox-hugo/temperature_sensor_pt100_curve.png" caption="Figure 7: 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 -R_meas_1 = 18:100; -R_meas_2 = 100:390; +%% "Perfect" temperature and resistance +R0 = 100; % [Ohm] +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; -plot(R_meas_2, T_meas) -xlabel('Resistance [$\Omega$]') -ylabel('Temperature [${}^oC$]'); +T1 = -200:dT:0; % [degC] +T2 = dT:dT:850; % [degC] +T_lut_linear = [T1,T2]; % [degC] +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 ``` + + +{{< figure src="/ox-hugo/temperature_sensor_lut_errors.png" caption="Figure 8: 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). + + +
+ Table 2: + Example of A, B, C and D coeficients for an NTC thermistor (DC95F202VN) +
+ +| | 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] +``` + + + +{{< figure src="/ox-hugo/temperature_sensor_ntc_curve.png" caption="Figure 9: 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) \\] + + +
+ Table 3: + Coefficients used to compute the temperature as a function of the resistance +
+ +| 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} diff --git a/static/ox-hugo/temperature_sensor_lut_errors.png b/static/ox-hugo/temperature_sensor_lut_errors.png new file mode 100644 index 0000000..dc0387f Binary files /dev/null and b/static/ox-hugo/temperature_sensor_lut_errors.png differ diff --git a/static/ox-hugo/temperature_sensor_ntc_curve.png b/static/ox-hugo/temperature_sensor_ntc_curve.png new file mode 100644 index 0000000..bfb8abc Binary files /dev/null and b/static/ox-hugo/temperature_sensor_ntc_curve.png differ diff --git a/static/ox-hugo/temperature_sensor_pt100_curve.png b/static/ox-hugo/temperature_sensor_pt100_curve.png new file mode 100644 index 0000000..76bdd18 Binary files /dev/null and b/static/ox-hugo/temperature_sensor_pt100_curve.png differ