Giles Henderson
and Ashvin Gajjar Eastern thois University Charleston, 61920
I
I I
Nonlinear h r ~ l n ~ l t c l l
M o s t instrumental designs strive to incorporate a linear display of the dependent variable. However, the amateur instrument builder is frequently faced with mechanical or financial circumstances that render this goal impractical. Frequently a calibration curve is invoked for nonlinear variables. Although this may sufficein terms of desired accuracy, it suffers several disadvantages and limitations. The preparation of a suitable calibartion curve is usually very timeconsuming and tedious. Further, if some instrumental parameter is changed, it may necessitate recalibration. These inconveniences may be efficientlyovercome by employing calibration equations. The general availability of digital computers particularly facilitates obtaining suitable equation parameters. Computer calculated parameters may be obtained in a relatively short time and may he recalculated to compensate for instrumental modifications or adjustments with minimum effort. Once a suitable calibration equation has been determined, the computer may either be used to routinely evaluate the calihration function, or if desired, the computer may print a calibration table. In addition to being faster and more flexible, the computer calibration technique is usually more accurate than graphical methods. Methods
Usually a specific function that describes the behavior of the variable is known and could he used as a calibration equation once suitable parameters are found. However, this approach usually requires an iterative procedure for a least-squares fit (1). Further, each different calibration function requires a different set of normal equations and appropriate computer program. An orthogonal polynomial equation offers several distinct advantages as a general calibration equation. Almost all nonlinear calibrations can be expressed to an excellent degree of accuracy by a single type of function. Hence a single computer program can he employed for several different calibration problems. The least squares solution for orthogonal polynominals can he carried out exactly without expanding the function in a Taylor series. Hence the usual iterative procedure employed in most nonlinear curve fitting procedures is avoided. The smaller number of calculations helps to minimize round-off error. The fitting procedure seeks to find the coefficients such that the orthogonal polynomial is a good fit of the calibration data in a least squares sense. Curve fitting programs of this type are readily available (g). In general, if an instrument designed to measure some variable of a chemical system, is found to give a
single-valued, nonlinear response, it may be calibrated in the following manner. Responses 6, (i = 1,2,. . .,m) are measured for m standards with known values Y, (i = 1,2. . .,m)of the variable under consideration. The variable may he expressed as a function of the response in the form of an nth order orthogonal polynomial equation The principle of least squares states that the best representation of the data is that which makes the sum of the squares of the differences between the observed values, Y,, and the function value, f(a1,. . .a,, a,), a minimum (i.e., a minimum regression, R). m
R
=
.
IYi - f(a1, . . a,, 6 i ) l 2 i=l
The least squares method requires that the numher of parameters be considerably smaller than the number of experimental values (i.e, n
