802,472. Electric analogue calculating systems. ELECTRIC & MUSICAL INDUSTRIES, Ltd. Jan. 13, 1955 [Jan. 27, 1954], No. 2451/54. Class 37. An interpolating device for deriving an analogue voltage accurately representing the displacement of a shaft or other movable element (e.g. in the automatic control of a machine tool) comprises a first voltage dividing transformer energized from a reference voltage source and having plural tapping-points from which corresponding selected voltages having different fractions of the reference voltage are derived by a multicontact selector switch operated in response to the input displacement, a second voltage dividing transformer of variable ratio also responsive to the input displacement from which is derived an incremental voltage which is a variable fraction of the voltage between a particular tapping of the first transformer and a neighbouring tapping thereof, and an electrical coupling between the outputs of the first and second transformers such that the incremental voltage and that selected from the first transformer are additively combined, whereby interpolation is effected between the voltages of the first transformer tapping points in response to the input displacement, and the second voltage dividing transformer is individually energized from the reference voltage source in such manner that non-uniform loading of the first voltage dividing transformer is avoided. In Fig. 1 a reference fixed alternating voltage excites a tapped autotransformer 1 having contacts a1 to a9 in two groups which are circularly arranged though linearly illustrated; the centre points of the contacts positionally representing successive discrete values of an independent input variable and the voltages thereon being analogous to the values of a function of the variable for the corresponding input values. Odd contacts are traversed by movable brush 25 and even contacts by brush 26 rotatable on a slow-speed shaft whose angular displacement continuously represents the independent variable. The brushes are connected to the midpoints of windings of autotransformers 6 and 7, semicircularly disposed about a high-speed shaft 9 geared to the shaft traversing brushes 25, 26 to make half a revolution for a displacement of the latter shaft corresponding to the separation between adjacent contacts; which traverses the brush 8 over windings 6, 7 to lie at the midpoint of transformer 6 when brush 25 is at the midpoint of any odd-numbered contact a1, a3, &c. and brush 26 is midway between any two even numbered contacts a2, a4, etc.; and to be at the midpoint of transformer 7 when brush 25 is midway between any two odd-numbered contacts a1, a3, &c. and brush 26 is at the midpoint of any even-numbered contact a2, a4, &c. Windings 6, 7 are respectively energized from equal independent windings 23, 24 of autotransformer 1, and the output voltage of brush 8 is developed across resistor 12, representing the sum of the discrete voltage derived by brush 25 (or 26) from the appropriate autotransformer tap in response to the slow-speed input shaft representing an independent variable, and an incremental signal representing the product of an interpolation coefficient (e.g. the constant first differential of the present function), and the difference (the independent increment) between the instantaneous value of the independent variable and the discrete value represented by the appropriate autotransformer tap. As the input shaft rotates in accordance with the independent variable, the incremental signal is maintained proportional to the required independent increment, while brush 8 makes before breaking and brushes 25, 26 break before making, so that a continuous signal is maintained and the input autotransformer is never non-uniformly loaded by autotransformers 6, 7. In a modification (Fig. 2, not shown) the high-speed shaft has two brushes spaced 180 degrees apart and rotatable over a single interpolating autotransformer winding energized by a single independent winding of the input autotransformer. The slow-speed brushes engaging the tappings of the latter are respectively connected to the brushes of the high-speed shaft, and the required output voltage is derived from a centre tap on the winding of the interpolating autotransformer. Fig. 3 shows a device for linear interpolation wherein the function of the independent variable is non-linear, the first derivative is not a constant, and the interpolation coefficient is again derived explicitly. Contacts a1, a2, a3-a9 arranged in two tracks derive from corresponding tappings of autotransformer 1 the voltage analogues of the required function in correspondence with equispaced discrete values of the input variable, while to correctly set up the function values there is inserted in series with each contact an auxiliary voltage developed from corresponding secondaries b1, b2, b3-b9 of a transformer 32 whose common primary 30 is energized from an independent winding 31 of the input autotransformer. Since the first derivative of the non-linear function is different at successive discrete values of the input variable, the interpolation coefficient voltages are inserted by an autotransformer 33 energized from independent winding 34 of the autotransformer 1, and tapped to supply graduated voltages combined with corresponding auxiliary correcting voltages from the windings of transformer 39 (similar to 32) to contacts c1, c2, c3 ... c9 arranged in two tracks and traversed by brushes 35, 36 ganged with brushes 25, 26 moving over contacts a1, a2, a3 ... a9 for operation by the low-speed input shaft rotated in accordance with the input variable. Brushes 35, 36 are coupled by transformers 37, 38 to energise interpolating autotransformer windings 6, 7 disposed semi-circularly on either side of high-speed shaft 9 geared to the low-speed shaft and carrying a brush 8 rotatable over the windings through 360 degrees to derive the output voltage, while brushes 25, 26 successively energize the midpoints of the windings with voltages derived from contacts a1, a2, a3 ... a9. Operation is similar to that of the device shown in Fig. 1 except that the energization voltage of the separately cored interpolating autotransformers is variable with rotation of the input shaft in accordance with the law of the first derivative of the function. Fig. 4 shows a device similar to that of Fig. 3 wherein incremental voltages corresponding to discrete values of the non-linear function are supplied to lineal contacts a1, a2, a3, &c. by tapped autotransformer 1 and multi-secondary transformer 32 energized from an independent winding 31 thereon. Superimposed upon these voltages are interpolation coefficient voltages derived from secondaries d2, d4, d6 &c. of a transformer 41 for contacts a2, a4, &c. and secondaries d1, d3, d5, &c. of transformer 42 for contacts al, a3, &c. which transformer primaries 43, 45 are energized respectively by 180 degrees spaced brushes 44, 46 driven by the high-speed shaft 9 geared to the low-speed input shaft over the semicircular winding of an autotransformer 47 energized by a centre-tapped winding 48 of autotransformer 1. Brush 40 traversed by the slow-speed shaft over contacts a1, a2, a3 &c. derives a continuously-variable voltage comprising the appropriate composite discrete analogue voltage plus the product of the first derivative of the function at each particular selected value of the input variable and the difference between such analogue voltage and that for the preceding value of the input variable. In Fig. 5 the voltages analogous to the discrete input function values are applied to contacts a1, a2, a3, &c. by parallel connections from tappings of autotransformers 49, 50 whose upper ends are connected to terminal 13 of an A.C. supply through transformer secondaries 51, 52 and whose lower ends are connected to terminal 14 of the supply through transformer secondaries 53, 54. Corresponding primaries 55, 58 are energized between the mid-point of an autotransformer 56 energized from the A.C. supply and the 180 degrees spaced brushes 57, 59 rotatable by high-speed shaft 9 geared to the low-speed input variable shaft over the semicircular autotransformer winding, to alternately excite tapped autotransformers 49, 50. The necessary incremental voltages are added to the discrete values of voltage applied to contacts a1, a2, a3, &c., by raising or lowering the energization voltage of the appropriate tapped autotransformer 49 or 50 in response to the position of the high-speed shaft 9 and its brushes 57, 59 traversing the winding of autotransformer 56, and the output voltage is derived from brush 40 traversed over contacts a1, a2, a3, &c. by the low-speed shaft. Fig. 6 shows a modification of Fig. 4 extended to an arbitrary function of two variables x, y, wherein the contacts of the slow-speed switch are arranged in a twodimensional matrix of rows x1, x2, x3, &c. and columns y1, y2, y3, &c. from which the required interpolated function voltage is derived by a brush 60 movable two-dimensionally over the matrix in accordance with the values of the input variables to embrace any four adjacent (x, y) contacts. Fixed tappings on autotransformer 1 energize the respective matrix contacts with voltages analogous to the function values for the discrete input values of (x, y) corresponding to the position of each contact, and there are provided interpolating transformers 61, 62 whose primaries are energized by 180 degrees spaced brushes 60, 67 driven by high-speed shaft 68 gearedly responsive to the X input variable over semicurcular autotransformer winding 65 independently energized from a secondary of autotransformer 1, and interpolating transformers 63, 64 whose primaries are similarly energized by brushes driven over a similar autotransformer winding by high-speed shaft 69 gearedly responsive to the Y input variable.