Patent GB-736085-A (GB)
Title: Improvements in or relating to electrical calculators
IPC: G06G7/36
Filed: 1952-07-28
Abstract: 736,085. Electric analogue calculating systems. SOC. D'ELECTRONIQUE ET D'AUTOMATISME. July 28, 1952 [July 26, 1951], No. 19043/52. Class 37. In an electric analogue computor for developing electric voltages representing the real and imaginary components of the polynomial having real coefficients a0, a1, a2, &c. and complex values of Z such that which is shown by algebra to be reducible to the real and imaginary components and for which it is also shown by algebra that where n is even; there are provided means for deriving from the primary variables x1, y1 two sets of variable unidirectional voltages representing the two sets of auxiliary variables x2, x4, x6, &c. and y2, y4, y6, etc.; means for multiplying the voltages representing the auxiliary variables by appropriate coefficients and for adding such multiplied voltages and appropriate unidirectional fixed reference voltages multiplied by other appropriate coefficients to produce voltages representing the sums #1 to #4 defined above, and further multiplying and adding means to combine such sums with each other and with the primary variables x1 and y1 in the manner defined above to derive voltages representing the real component and the imaginary component of the polynomial Z = P(z). Figs. 1, 2 show a calculator wherein the primary variables x1, y1 derived from equation (2) control the rotation of shafts 1, 23 to adjust the sliders of ganged linear potentiometers 7 to 11 and 28 to 31 respectively. Potentiometer 7 is energized by a reference voltage Œv and the slider output is mixed with the input analogue voltage x1 at the input of amplifier 14 exciting a servomotor 12, which rotates shaft 1 proportionately to x1 until the amplifier input is zeroized. Potentiometer 8 energized from the input by voltage - x1 through a retroactive phase reversing amplifier 19 and by voltage +x1 direct develops a slider voltage x1<SP>2>. Shaft 23 is similarly rotated proportionately to y1 by servomotor 32 energized from the amplified difference between the input analogue voltage y1 and the slider output of potentiometer 28 which is energized by a reference voltage ŒV to zeroize the amplifier output and the slider output of potentiometer 29 energized with voltages #y1 develops a voltage - y1<SP>2>. A similar self-balancing amplifier-servomotor combination energized by voltages x1<SP>2> and - y1<SP>2> rotates shaft 50 controlling potentiometers 56 to 60 and potentiometer 56 energized by a reference voltage ŒV develops a rebalancing slider voltage - x2=x1<SP>2> - y1<SP>2> which is fed back to the amplifier input so that the shaft rotation is proportional to x2 and potentiometer 57 energized by voltages #x2 develops on its slider a voltage x2<SP>2>. Potentiometer 9 energized by voltages #y1 develops a slider voltage x1, y1 energizing a self-balancing-amplifier-servomotor combination together with the slider output of potentiometer 84 to rotate shaft 79 driving potentiometers 84 to 87 proportionately to y2=2x1y1 so that the slider voltage of potentiometer 84 energized with a reference voltage ŒV zeroizes the amplifier input and after phase reversal energizes potentiometers 58, 85 respectively with voltages #y2 and Œy2 so that their sliders develop voltages proportional to - 2x2y2= - y4 and +y2<SP>2>. The outputs of potentiometers 57 and 85 are mixed in retroactive amplifier 104 to a voltage x4=x2<SP>2> - Y2<SP>2> on terminals 106, 107 which is phase reversed in amplifier 109 to a voltage - x4 on terminals 110, 112, while the output of potentiometer 58 is phase-reversed to a voltage y4 on terminals 116, 118 and again phase-reversed to a voltage - y4 on terminals 121, 123. Voltages Œx2 and Œy2 also appear on terminals 66, 71 and 92, 97 respectively. Potentiometer 59 (Fig. 2) driven by shaft 50 is energized by voltages #x4 on terminals 112, 107 to produce a slider voltage x2, x4 and potentiometer 86 driven by shaft 79 is energized by voltages Œy4 on terminals 118, 123 to produce a slider voltage - y2 y4. The outputs of the latter two potentiometers are added at the input of the retroactive amplifier 134 to produce an output voltage x6=x2x4 - y2y4 on terminal 135 and a phase reversed voltage - x6 on terminal 139. Similarly, potentiometer 60 driven by shaft 50 is energized by voltages Œy4 on terminals 118, 123 to produce a slider voltage +x2y4 while potentiometer 87 driven by shaft 79 is energized by voltages Œx4 on terminals 107, 112 to produce a slider voltage +x4y2 and the voltages are added at the input of retroactive amplifier 143 to produce a voltage - y6= - (y2x4+x2x4) on terminal 144 which is phase reversed in amplifier 149 to +y6 on terminal 150. Further voltages +x8+y8, Œx10Œy10, &c. may, if necessary, be developed analogously, using further cascaded potentiometers driven from the shafts 50 and 79 and energized by the voltages developed above. Function #2 is synthesized by addition at the input of retroactive amplifier 157 whose input is derived from potentiometers 152, 153, 154, 155 respectively energized by a reference voltage ŒV1 and by voltages Œx2, Œx4 and Œx6, the potentiometer sliders being manually adjustable to set in the coefficients a1, a3, as, a7 and the combined output voltage representing #2 in accordance with equation (6) above. Function #4 is similarly synthesised in retroactive amplifier 166 fed from the sliders (manually adjusted in accordance with coefficients a3' a5, a7) of potentiometers respectively energized by voltages Œy2, Œy4 and Œy6, in accordance with equation (8) above. The E2, #4 voltages are respectively phase reversed in amplifiers 159, 168 and potentiometers 10, 11 driven by shaft 1 are respectively energized by voltages ##2 and ##4 while potentiometers 30, 31 driven by shaft 23 are similarly energized by voltages Œ #4 and ##2. The outputs of potentiometers 10, 30 representing -x1#2 and y1 #4 are mixed in amplifier 178 with the outputs of potentiometers 179, 180, 181, 182 respectively energized by voltages #V, #x2, #x4 and #x6 whose sliders are set manually in accordance with coefficients a0, a2, a4, a. to represent #1 according to equation (5). The output of amplifier 178 on terminal 183 represents the real components of the polynomial given by The outputs of potentiometers 11, 31 representing -x1#4 and - y1#2 are similarly mixed in amplifier 187 with the outputs of potentiometers 188, 189, 190 respectively energized by voltages #y2, #y4 and #y6, where sliders are set manually in accordance, with coefficients a2, a4, a, to represent #3 according to equation (7). The output of amplifier 187 on terminal .191 therefore represents the imaginary component of the polynomial given by To solve the polynomial equation z=p(z)=0 for its real and imaginary roots, i.e. to determine the values for which where # is an arbitrary small quantity, the voltages representing R(z) and P(z) on terminals 183, 191 (Fig. 5), are amplified and supplied to bridge rectifiers 233, 234 whose positive outputs are mixed at the input of a common retroactive D.C. amplifier 235, whose negative output represents the modulus of Z as the primary variables x1, y1 are varied according to a predetermined law and which is applied to the suppressor grid of a pentode 236, in opposition to a small positive bias 243 which with a standing control grid positive bias normally holds the tube blocked when the negative output of the amplifier is applied but unblocked when the latter is reduced to a value corresponding to the approximation value e so as to negatively trigger a monostable flip-flop circuit 237, which positively pulses the control grid of a cathoderay tube 239 whose x and y deflection coils are energized proportionately to the primary variables x1 and y1 so that the spot position indicates the values at which the modulus of Z lies within the desired approximation to zero. The coefficients of the polynomial are set up manually and x1 is varied, leaving y1 fixed. The real roots of the polynomial for the fixed value of y1 are given by the visible C.R.O. trace, and the process is repeated for different values of y1 whereby the field of real and imaginary roots is explored for a range of values of x1 and y1. The exploration may be carried out by automatic regulation of x1 and y1. In a modification (Fig. 3) variables a, b such that are mechanically set in to input shafts 192, 205, and it is shown that The x1, y1 rotations of shafts 1, 23 are derived from shafts 192, 205 through trigonometrical calculating mechanisms 193, 194; 203, 204 in accordance with equation (15) above, while shafts 206, 207, geared to shafts 192, 205 drive differentials 208 in similar directions and 209 in opposite directions to drive shafts 210, 211 in accordance with (a + b) and (a - b) and drive shafts 215, 216 through trigonometrical calculating mechanisms 213, 214 in accordance with cos (a+b), cos (a - b) respectively. The shafts drive the differential 217 in opposite directions so that its output shaft rotate shafts 79 in accordance with y2 according to equation (17) above. The x1, y1 rotations of shafts 1, 23 operate further trigonometrical calculating mechanisms 195, 202 to drive shafts 196, 200 proportionately to 2x1<SP>2>=(1 - cos 2a) and 2y1<SP>2>=(1- cos 2b), and the rotations are oppositely combined in differential 197 to drive shaft 50 through reduction gear 199 proportionally to x2 according to equation (16) above. The trigonometrical calculating mechanisms may comprise (Fig. 4) equal rotary discs 219, 221 centred by an eccentric cross-bar 222 slidable through a block 223 movable between guides 224, 225 at right-angles to cross-bar 222 and fixed to a belt 230 carried by rollers 227, 229. The rotation applied to the input shaft 218 of disc 219 displaces the block 223 vertically up or down and belt 230 thus drives roller 229 and output shaft 228 through a sine function of the input rotation. If the input rotation is applied to shaft 220 of disc 221 the