Analytic form of transfer function. In certain cases the transfer function is available as an analytic expression. One common transfer function used for resistance temperature sensors (to be discussed in Chapter 3) is the Callendar– Van Duzen equation. It gives the resistance of the sensor at a temperature T as
$$R(T)=R_{0}(1+AT+BT^2+C(T-100)T^3) \enspace,$$where the constants A, B, and C are determined by direct measurement of resistance for the specific material used in the sensor and $R_0$ is the temperature of the sensor at 0 ºC. Typical temperatures used for calibration are the oxygen point (-182.962 ºC; the equilibrium between liquid oxygen and its vapor), the triple point of water (0.01 ºC; the point of equilibrium temperature between ice, liquid water, and water vapor), the steam point (100 ºC; the equilibrium point between water and vapor), the zinc point (419.58 ºC; the equilibrium point between solid and liquid zinc), the silver point (961.93 ºC), and the gold point (1064.43 ºC), as well as others. Consider a platinum resistance sensor with a nominal resistance of 25 $\Omega $ at 0 C. To calibrate the sensor its resistance is measured at the oxygen point as 6.2 $\Omega $, at the steam point as 35.6 $\Omega $, and at the zinc point as 66.1 $\Omega $. Calculate the coefficients A, B, and C and plot the transfer function between -200 ºC and 600 ºC.
In order to obtain the sensor calibration, several measurements at different temperatures where taken:
6.2 $\Omega $ at a temperature of -182.962 ºC (oxygen point).
35.6 $\Omega $ at a temperature of 100 ºC (steam point).
66.1 $\Omega $ at a temperature of 419.58 ºC (zinc point)
Let's plot the points,
In [10]:
import matplotlib.pyplot as plt
import numpy as np
from math import log, exp
%matplotlib inline
from scipy.interpolate import InterpolatedUnivariateSpline
T_exp= np.array([-182.962,100,419.58]);# Celcius
R_exp= np.array([6.2 ,35.6,66.1])# Ohm
plt.plot(T_exp,R_exp,'*');
plt.ylabel('Resistance of the sensor [Ohm]')
plt.xlabel('Temperature [C]')
plt.show()
Reordering the Callendar-Van Duzen equation we obtain the following
$$ AT+BT^2+C(T-100)T^3 =\frac{R(T)}{R_0}-1 \enspace,$$which we can write in matrix form as $Mx=p$, where
$$\begin{bmatrix} T_1 & T_1^2 & (T_1-100)T_1^3 \\ T_2 & T_2^2 & (T_2-100)T_2^3 \\ T_3 & T_3^2 & (T_3-100)T_3^3\end{bmatrix} \begin{bmatrix} A\\ B \\ C\end{bmatrix} = \begin{bmatrix} \frac{R(T_1)}{R_0}-1 \\ \frac{R(T_2)}{R_0}-1 \\ \frac{R(T_3)}{R_0}-1\end{bmatrix} \enspace.$$Because $M$ is square we can solve by computing $M^{-1}$ directly.
In [11]:
R0=25;
M=np.array([[T_exp[0],(T_exp[0])**2,(T_exp[0]-100)*(T_exp[0])**3],[T_exp[1],(T_exp[1])**2,(T_exp[1]-100)*(T_exp[1])**3],[T_exp[2],(T_exp[2])**2,(T_exp[2]-100)*(T_exp[2])**3]]);
p=np.array([[(R_exp[0]/R0)-1],[(R_exp[1]/R0)-1],[(R_exp[2]/R0)-1]]);
x = np.linalg.solve(M,p) #solve linear equations system
np.set_printoptions(precision=3)
print('M')
print(M)
print('\n')
print('p')
print(p)
print('\n')
print('x')
print(x)
We have found the coeffiecients $A$, $B$, and $C$ necessary to describe the sensor's transfer function. Now we plot it from -200 C a 600 C.
In [5]:
A=x[0];B=x[1];C=x[2];
T_range= np.arange(start = -200, stop = 601, step = 1);
R_funT= R0*(1+A[0]*T_range+B[0]*(T_range)**2+C[0]*(T_range-100)*(T_range)**3);
plt.plot(T_range,R_funT,T_exp[0],R_exp[0],'ro',T_exp[1],R_exp[1],'ro',T_exp[2],R_exp[2],'ro');
plt.ylabel('Sensor resistance [Ohm]')
plt.xlabel('Temperature [C]')
plt.show()
We see the fit is accurate. Note that our approach is also valid if we have more experimental points, in which case the system of equations $Mx=p$ is solved in the Least-Squares sense.
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