Patent GB-849091-A (GB)
Title: Improvements relating to apparatus for deriving a differential coefficient
Inventor: SPENCER ROLF EDMUND
IPC: G05B19/4105
Filed: 1955-12-15
Abstract: 849,091. Electric analog calculating. ELECTRIC & MUSICAL INDUSTRIES Ltd. Dec. 5, 1956 [Dec. 15, 1955], No. 35983/55. Class 37 Apparatus for obtaining the first derivative of a curvilinear function of a variable comprises at least three input terminals, means responsive to signals applied to said input terminals and representing ordinates of reference points which define said function, for deriving signals representing respectively the first and succeeding derivatives of the function at a particular value of the variable, and means for combining said derived signals in such proportions as to produce an output signal substantially representing the first derivative of the function at one or more other values of the variable. In Fig. 1, a switching device 1 of the kind described in Specification 796,994 continuously selects from a record successive groups of three cophased alternating voltages representing values y 1, Y 2, y3 3 of a dependent variable y for equally spaced values of an independent variable x, of which y 1 , y 3  are fed to the ends A, B of a tapped transformer primary 2 and y 2  is fed to the end D of a further transformer primary 3 returned to the midpoint C of primary 2, the latter having graduated tappings respectively connected over graduated secondaries 6 coupled to primary 3 to a range of switch contacts 4 semicircularly disposed about a contact connected to D, and swept by a mechanically driven moving contact 5 in synchronism with the switching device 1 to pick off voltage values of y interpolated over the range y1, y z , y  3  in accordance with the position of the switch drive shaft continuously responsive to x; a further set of three values y y2 Y 3 being selected by switch I for each traverse of the moving contact and the primary windings 2, 3 having three degrees of freedom-viz. the magnetic flux linking 2, the voltage at the midpoint thereof, and the magnetic flux linking, 3 constrained by the voltages at A, B, D. A secondary 7 coupled to primary 2 is returned to earth and connected to the centrepoint of secondary 8 coupled to primary 3, whose ends energize potential divider 9 of a linear interpolator (which may be of the kind set forth in Specification 802,472) equally tapped to plural semicircularly disposed contacts corresponding to contacts 4, traversed by a moving contact 12 ganged to contact 5 to develop a signal 13. It is shown (Fig. 2) that the voltages at the primary taps of transformer 2 represent values of y disposed along ' a straight line joining points P 1   (x 1 , y 1 ) and P 3  (x 3 , y 3 ) which together with the intermediate point P 2  (x 2 , y z ) lie on a quadratic curve given by: y = f(x) = ax<SP>2</SP> + bx + c and by mathematics that the input voltage at AB across the primary of transformer 2 is proportional.to the first derivative dy of y with resdx pect to x at point P 2 , so that the voltage developed by secondary winding, and thus at the midpoint of 8 applied to the potential divider 9 is similarly proportional; while the voltage across primary 3 and therefore across secondary 8 is proportional to the second derivative d<SP>2</SP> y dx<SP>2</SP> (which is constant since y is a quadratic function of x) and to interval P 2 R between the y coordinates of points P 2  and R (Fig. 2) at x =  X2 , so that the total voltage across secondary 8 completely represents the first derivative of y at P 2 , and may be linearly interpolated for values of x by potential divider 9 and moving contact 12. The relative inductance values of windings 7, 8 are adjusted so that their outputs are to the same scale, and it is shown that the interpolated output at point 13 represents the value of the 1st differential coefficient of the corresponding interpolated value of y obtained by moving contact 5; linear interpolation being sufficient since the curve of variation of the first differential coefficient over successive values of (x  1 , y1) (x 2 , y 2 ) and (x 3 , 3, y 3) represented by points Q  1 , Q 2 , Q  3  (Fig. 3) is linear; the point Q 2  representing the differential coefficient at x = x 2 . The inductance 8 may be tapped directly to switch contacts 11 to form the potential divider. If points P 1 , Pz, P 3  are collinear, the inductance 3 may be short-circuited to eliminate the second differential coefficient signal. Sub-interpolation of the output signals from moving switch contacts 5 and 12 may be employed, and the apparatus may be duplicated for each unit to be operated on alternate sets of three y - input signals, which are successively interchanged. The apparatus may be operated in response to a time variable e.g. in conjuction with a parametric interpolator as set forth in Specification 830,462, and is applicable to the control of automatic machine tools. Specifications 848,411 and 848,412 also are referred to.