9. APPENDIX

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The appendix discusses the computer programs used for the optimization of the temperature characteristics of the zero-drift and span-drift compensation bridges. The programs were written in Fortran and run on an ODRA 1305 computer.¹

9.1. Description of the program optimizing the characteristics of the zero-drift compensation bridge

The characteristics of the zero bridge were optimized with a view to minimizing the slope of the nonlinearity function of the resistor-thermistor divider (Fig. 32) present in each arm of the zero-drift compensation bridge. This divider has two degrees of freedom. One of them, the value of the resistor $R_{1}$, was used to obtain a division ratio equal to 0.5 (zero voltage across potentiometer P) at the reference temperature $T_{3}$. The second degree of freedom, in the form of the value of the resistor $R_{2}$, was used to minimize the slope of the nonlinearity function.

All printed results for the nonlinearity function and the slope of the nonlinearity were referred to the span of the transducer. This normalization was carried out by assuming that the change of the division ratio between the temperatures 253 K and 333 K corresponds to 8 % of the span.

The input data to the program were the resistance values of the thermistors at the temperatures $T_{1}$ = 253 K, $T_{2}$ = 273 K, $T_{4}$ = 293 K, $T_{6}$ = 313 K, $T_{7}$ = 333 K for the thermistor of maximum and of minimum resistance in the batch under study, and the resistance values of a hypothetical thermistor whose resistance values are equal to the mean resistance of the thermistors at each temperature.²

The simplified computation procedure was as follows:

1. Assume $T_{3}$ = 288 K, $T_{5}$ = 298 K, $R_{2}$ = 59 Ω, for the mean thermistor characteristic

2. Compute $R_{1}$ such that the division ratio WD = 0.5 at the temperature T = $T_{3}$

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3. For this $R_{1}$ compute WD at the remaining six temperatures

4. Compute, at all seven temperatures, the ordinates of the straight line having the same values as the ratio WD at the temperatures $T_{3}$ and $T_{5}$

5. Compute the differences D between the ratio WD and the ordinates of the straight line from step 4 at the seven temperatures

6. Compute the coefficients of the equations of the straight lines having values in common with the difference D at the temperature 293 K and at each of the remaining six temperatures

7. Find and store the maximum value M of the modulus of the slope of the six straight lines from step 6

8. Repeat the computations of steps 2 to 7 150 times, changing the value of $R_{2}$ by +1 % at each step

9. Compute the minimum value of M in the set of 150 values. For this value print the computed values of the resistors $R_{1}$ and $R_{2}$.

10. For the values of $R_{1}$ and $R_{2}$ found, compute and print the difference D at the seven temperatures for all three thermistor characteristics, print the range of variation of the ratio WD between the temperatures 253 K and 333 K, and the value of the modulus M.

11. Compute and print the value of the maximum difference D for ten values of the resistor $R_{2}$ smaller than the optimum and for ten values of the resistor $R_{2}$ larger than the optimum, changing the resistor value by 5 Ω each time.

12. Repeat computations 2 to 11 for the remaining seven pairs of temperatures $T_{o}$ – $T_{1}$.³

13. Print the maximum value of D from step 10 (the nonlinearity) as a function of the temperature pairs $T_{3}$ – $T_{5}$.

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9.2. Description of the program optimizing the characteristics of the span-drift compensation bridge

The linearization of the voltage $U_{AC_{1}}$ (Fig. 35) as a function of temperature was carried out by a computer procedure analogous to that for the zero-drift compensation bridge. The differences lay in a different initial condition – select the resistor $R_{1}$ so that, for the assumed $R_{3}$, $R_{5}$, a relative change of the voltage $U_{AC}$ referred to the temperature 293 K equal to 20 % is obtained – and in a different procedure for searching for the extremum. Since, after the value of $R_{1}$ had been fixed, two degrees of freedom still remained – the values of the resistors $R_{3}$, $R_{5}$ – four search-density levels were introduced. At the first level the program searched for the minimum slope of the nonlinearity for eight values of each of the resistors, varied with a ratio of 1.64, starting from 200. At the second level, the search for the minimum slope began from the value preceding the optimum value found for each resistor, and was likewise carried out for a further eight values with a ratio of 1.16. At the third level the ratio was 1.04 and at the fourth, last level 1.01.⁴

As a result of the search, at each level the minimum was sought from a set of 84 values.⁵

For each of the pairs of temperatures $T_{3}$ – $T_{5}$ the computed optimum resistance values $R_{1}$, $R_{3}$, $R_{5}$ were printed, together with the nonlinearity as a function of temperature for these values for the three thermistor characteristics. In addition, the nonlinearity was printed for 10 values smaller and 10 values larger than the computed optimum values of the resistors $R_{3}$ and $R_{5}$, as well as the maximum slope of the nonlinearity characteristic as a function of the temperature pairs $T_{3}$ – $T_{5}$.

The normalization of the results was carried out taking into account that the change of the division ratio between the temperatures 253 K and 333 K corresponds to 20 % of the measuring range.