Apparatus for Continuous Differentiation of Data CurvesC. H. Chervenka Citation: Review of Scientific Instruments 37, 553 (1966); doi: 10.1063/1.1720249 View online: http://dx.doi.org/10.1063/1.1720249 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/37/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extracting dispersion curves of acoustic data with continuous wavelet transform J. Acoust. Soc. Am. 123, 3373 (2008); 10.1121/1.2933995 Continuous differential signal equalizer J. Acoust. Soc. Am. 90, 1217 (1991); 10.1121/1.401984 Apparatus for ``Hyperbolic Glow Curves'' Rev. Sci. Instrum. 33, 1168 (1962); 10.1063/1.1717722 On the Use of Curve Differentials in Thermodynamics Am. J. Phys. 19, 284 (1951); 10.1119/1.1932804 An Instrument for Mechanically Differentiating Curves Rev. Sci. Instrum. 21, 397 (1950); 10.1063/1.1745594

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

141.209.100.60 On: Sun, 21 Dec 2014 18:33:23

THE REVIEW OF SCIEKTIFIC INSTRUMENTS VOLUME 37. NUMBER 5 MAY 1966

Apparatus for Continuous Differentiation of Data Curves

C. H. CHERVENKA

Spinco Division oj Beckman Instruments, Inc., 1117 Calijornia Avenue, Palo Alto, California 94304

(Received 13 December 1965)

A new mechanical device is described which rapidly yields the derivative of a data curve in the form of a con­tinuous plot. This table top differentiator, the operation of which is mostly manual, depends in principle upon the measurement of the slope of successive tangents to the curve. Slopes of numerical values between -1 and 5 are measured conveniently. The device has been applied to the differentiation of interference patterns from the analyti­cal ultracentrifuge.

INTRODUCTION

THE planimeter is well-known in scientific laboratories as a simple, effective device for the rapid integra­

tion of data curves, but there does not seem to be an analogous device readily available for the differentiation of data curves. While several differentiators bave been de­scribed in the literature, these are in general gauges or optical devices to replace or supplement the protractor in the measurement of slopes at a point on a curve. I •2 A device described by Rarbou in 19303 allowed the continuous inte­gration or differentiation of data curves, but the versatility of the apparatus limited its effectiveness for differentiation. For example, the accurate recording of the differential plot depended entirely upon the sidewise movement of a di­agonal wheel upon the chart paper. Measurements were limited to curves with slopes of maximum numerical values of less than one. Another reversed integrator was described by Scott,4 but this device did not yield a continuous dif­ferential curve, and also was limited unduly in the maxi­mum slope measurable.

This communication describes a relatively simple device, called the slope plotter, for the rapid differentiation of experimental data in the form of y vs x graphs. The slope plotter yields continuous curves of dy/dx vs x, in which the scale of the x axis is the same as that of the original graphs. The maximum slopes, positive or negative, which can be measured are set (except for slopes approaching infinity) only by the physical size of the apparatus and by the sensitivity desired. The device described at present is effective for slopes up to 5 (79°); higher slopes can be measured after simple modification of the apparatus. Negative slopes of -1 can be measured with the present apparatus.

Differentiation with the slope plotter is based in principle upon the matching of a scribe line to the tangent of the function curve. The inherent uncertainty existing in this procedure is minimized by the continuous plotting feature making available an almost infinite number of points in

1 T. Yamamoto, Bull. lost. Phys. Chem. Res. 11, 761 (1932). 2 M. S. Sytilin, Russ. J. Phys. Chem. 34, 660 (1960). • E V. Harbou, Z. Angew. Math. Mech. 10, 563 (1930). • A. H. Scott, Rev. Sci. lostr. 21, 397 (1950).

553

the differential curve. The slope plotter in principle yields the true derivative of the function curve. We have avoided the use of special optical devices to find the tangent of the curve, because for real function curves these do not seem to be better than a simple line, and in general they are not well suited for the continuous matching of changing slopes.

DESCRIPTION OF APPARATUS

The slope plotter is represented schematically in Fig. 1. Carriage (1) moves laterally on baseboard (3), being driven by a variable speed dc motor (4) through a gear and rack assembly. Carriage (2) moves up and down in a recess in baseboard (3). Rod (5) slides up and down in bearings (6) and (7) attached to (1). Ring (8) contains a transparent plastic disk with a long horizontal scribe line and a short vertical scribe mark and rotates in a hole in (1). Bearing (9) is attached to rod (5) by a pivot and moves with (5), in turn rotating ring (8) by sliding along rod (10), which is attached to (8). Wire (14) is attached to rod (5) and moves chart pen (11) as (5) moves.

In use, the y vs x data plot (12) is mounted on carriage (2). As carriage (1) is driven laterally, the operator moves carriage (2) up or down until the scribe mark on the plastic disk in (8) falls on the curve of the data plot, and simul­taneously moves rod (5) up or down until the scribe line

Fw. 1. Arschematic of the siopel'Piotter.

3

13

12

2 -, I

~

\ 11

7 r--

7 -51;

k 8G 6 =-

'\...4~

10 eM

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

141.209.100.60 On: Sun, 21 Dec 2014 18:33:23

554 C. H. CHERVENKA

lies along the tangent to the curve at the point. As the lateral carriage moves, these motions are continued and the relationships maintained continuously until the entire data curve is traversed. Meanwhile, pen (11) draws a curve on chart (13), attached to (3), which is a plot of dy/ dx vs x. The scale of the x ax:is of this plot is the same as that of the original function curve.

The sensitivity, that is the relationship between the slope of the data curve and the magnitude of dyjdx on chart (13) is variable by adjustment of the distance be­tween the pivot on bearing (9) and the center of (8). Two sensitivities have been used: low, in which the pen moves 2.60 cm for a slope of unity, and high, in which the pen moves 7.98 em per unit slope. Slopes higher than 5 can be measured either by decreasing the sensitivity still fur­ther, or by increasing the lengths of rods (5) and (10) and increasing the distance between ring (8) and bearing (7). The lateral carriage speed used most often has been approximately 4 cm/min.

PROCEDURE

The principal application of the slope plotter so far has been to the differentiation of interferograms from the ana­lytical ultracentrifuge. The finished glass plates from the ultracentrifuge were enlarged photographically on double weight, high contrast paper so that the final image was about 26 cm between reference edges; this corresponds to a total magnification of about 16X the actual sample cell dimensions. A line parallel to the fringes in the reference hole images was ruled across the pattern to serve as a baseline of zero slope. The enlargement was placed in the slope plotter and scanned directly; however, since no single fringe traversed the entire pattern, a procedure slightly different from that described above was used for scanning. A dear fringe was selected at the meniscus and scanned

FIG. 2. Typical results obtained with the slope plotter. IIigh sensi­tivity was used. The derivative plot is shown at the upper right with three consecutive traces superimposed. The top of the rectangle on the left is a calibration mark for a slope of 1. The interferogram from which the plot was derived is shown in the lower half of the figure.

until it ran out, then carriage (2) was shifted to a fringe lower in the pattern and this scanned until it ran out. This process was repeated until the entire fringe pattern was traversed. The zero slope line was then scanned to establish a baseline; the positions of the meniscus and bottom of the fluid column in the centrifuge cell were marked by moving rod (5) up and down to make a vertical line on the graph then the lateral carriage was in the proper position. An alternate procedure of scanning the interferograms has been used also; in this case, carriage (2) was positioned so the fringe pattern was centered verti­cally under the platsic disk, then allowed to remain in this position. As carriage (1) was driven laterally, the scribe line on the plastic disk was matched to the tangent of each fringe as is passed under the scribe mark.

RESULTS AND DISCUSSION

In Fig. 2, an example of a dy/dx vs x plot drawn using the slope plotter is aligned with the interferogram from which it was derived. Three traces are superimposed to illustrate the typical variations encountered. A limited statistical analysis of the results of repetitive scans is given in Table 1.

TABLE I. Reproducibility of the slope plotter.

Corresponding dy/dxa Mean value Standard position in cell Range of values N=21 deviation

Meniscus 2.68-3.12 2.89 0.146 ! level 4.05-4.30 4.18 0.066 j level 5.91-6.13 6.05 0.065 Bottom 8.54--9.30 9.04 0.211

a High sensitivity used.

The largest source of error in the operation of the slope plotter is in the matching of the scribe line to the tangent of the curve. Uncertainties result partially due to the nature of the line making up the experimental curve; the line has finite width and may have small transient vari­ations in slope, such as noise in recorder traces. (Optically formed curves such as the interference pattern illustrated in Fig. 2 are remarkably smooth, however.) The operation of the slope plotter is sufficiently convenient so that multi­ple scans are easily made, and the results can be averaged to reduce errors. We have routinely superimposed and averaged three traces. This procedure also minimizes the effect of another problem which arises-the relatively large deviations in the dy/dx plot due to momentary lack of attention on the part of the operator. The decreased pre­cision at the ends of the fluid column indicated by the re­suIts in Table I undoubtedly reflects the difficulty of matching the tangent of a curve when the curve ends at the point of tangency.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

141.209.100.60 On: Sun, 21 Dec 2014 18:33:23

DIFFERENTIATION OF CURVES 555

It is obvious that the operation of the slope plotter is highly subjective; however, with moderate experience the average operator should be able to reproduce results with precision of the order of that indicated in Table I. No attempt has been made to compare the results obtained with the slope plotter to those obtained by graphical differ-

THE REVIEW OF SCIENTIFIC INSTRUMENTS

entiation, since the results of the latter procedure are highly dependent upon the amount of effort exerted in the process. It is expected, however, that the results obtained with the slope plotter will prove to be as accurate as those arrived at by a moderately extensive graphical differentia­tion, but with far greater ease and convenience.

VOLUME 37. NUMBER 5 MAY 1966

Alternating Gradient Electrostatic Accelerating Tube D. BoYD*

Rutgers, The State University, New Brunswick, New Jersey 08900

AND

J. V. KANEt

Bell Telephone Laboratories, Murray Hill, New Jersey 07971

(Received 12 November 1965)

The processes of electron multiplication and electron loading in acceleration tubes have been investigated theoretically. These effects limit the maximum voltage at which Van de Graaff generators can operate. A new design for an accelerating tube is proposed which by virtue of its closely spaced electrodes has the advantage of ease of calculation and flexibility of boundary condition. Computer calculations were done for this tube to study electron multiplication trajectories. We have found that an axial alternating electrical field gradient can limit such ava­lanches by periodically trapping the electrons.

1. INTRODUCTION

RECENT advances in Van de Graaff technology have resulted in machines which can operate at voltages

in excess of 10 Me V and produce particle beams of energies in excess of 20 MeV by utilization of the tandem principle. The limitations in utility of this machine are due primarily to the failure of the insulating properties of the accelerator tube. Many existing Van de Graaffs could operate at higher voltages if a more satisfactory accelerating tube were available.

The two types of discharge that are found to occur are "electron loading" which results in a steady electron cur­rent down the tube, and a more violent form of this effect which results in violent tube spark or "kick." Both of these effects have been reduced recently with the advent of the inclined field tube, which "sweeps" electrons out of the beam.1 Unfortunately, it is difficult to calculate the elec­tric field due to inclined field electrodes, and the effect can only be estimated, or investigated experimentally. The same disadvantage also holds for other special devices such as various magnetic field configurations. These problems make the focusing and control of beam trajectories energy dependent and difficult to predict.

* Supported in part by the National Science Foundation. t Present address: Cylotron Bldg., Michigan State University, East

Lansing, Michigan 48823. 1 J. G. Trump, Nucl. Instr. Methods 28, 10 (1964); and K. H.

Purser, A. Galejs, P. H. Rose, R. J. Van de Graaff, and A. B. Witt­kower, Rev. Sci. Instr. 36, 453 (1965).

We believe that it is possible to construct a tube more suitable for calculation. Such a tube would have numerous electrodes, closely spaced, perhaps several to each 2.5 cm, so that to a good approximation a uniform boundary con­dition is obtained. Figure 1 shows how such a tube might be constructed and how the resistors could be attached. Pairs of resistors are suggested in order to reduce the chance of producing an open circuit in the expectation of resistor failures. By means of a suitable resistance divider network, the boundary potential shown in Fig. 2 could be produced. Calculations based on this trapping tube were done and

o • '--'

eM

FIG. 1. Electrode structure. The resistors are wound about the cir­cumference and connected to tabs protruding from the electrodes. In the ~onfiguration shown, the resistors would give rise to a linear gradIent on the tube. Various network schemes are possible which would lead to the oscillating potential of Fig. 2.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

141.209.100.60 On: Sun, 21 Dec 2014 18:33:23