Patent GB-1173900-A (GB)
Title: Improved Apparatus for the Solution of Mathematical Problems by Analogue Technique
Inventor: WALTON JOHN HUGH DAVEY
IPC: G06G7/122
Filed: 1966-12-21
Abstract: 1,173,900. Analogue solution of simultaneous equations and irregularities. J. H. D. WALTON. 21 Dec., 1966, No. 57168/66. Heading G4G. Apparatus for simultaneous solution of combined sets of equations or inequalities comprises an electrical impedance network of elements respectively coupling plural row conductors to plural column conductors; the elements being of impedances proportional to corresponding coefficients or constants and there being provided sources applying independently variable voltages to the several column conductors with at least one voltage opposed in phase or polarity to the remainder, and measuring means determining the magnitude and phase of voltage on each row conductor with respect to a common reference; so that the independently variable voltages are proportional to the variables of the equations or inequalities and the measuring means indicate satisfaction or otherwise of the equations or inequalities for the set in variables; such trial values converging to a solution when all such equations or inequalities are satisfied. Fig. 1 shows apparatus for solving simultaneous equations in which sets of resistance elements 1 to 6 and 21 to 23 are connected between row conductors 9, 10, 11 and column conductors 7, 8, 12 so that each column is connected to each row over a resistance element; battery 13 positively energizing variable potentiometers 15, 16 and voltmeters 24, 25 to energize columns 7, 8 with variable voltages v 1 , v 2  while battery 14 negatively energizes column 12 with voltage - V. Current meters 17, 18, 19 are connected in series with the respective rows and battery earth, and read i 1 , i 2 , i 3 . If the reciprocals of resistances 1, 2, 4, 5 are given by a 1,1 ; a 1,2 ; a 2,1 ; a 2,2 ; and those of resistances 3, 6 by b 1 ; b 2 ; while g 1 , g 2  denote the conductances of meters 17, 18, it is shown that where less than 1 and greater than zero; and for zero current in meter 17. so that if the conductances a 1,1 ; a 1,2 ; a 2,1 ; a 2,2 ; b 1 , b 2 ; represent the coefficients of two simultaneous linear equations and v 1 , v 2  are adjusted to zeroise i 1 , i 2 , the unknowns are represented by v 1 , v 2  inducted by voltmeters 24, 25. In evaluation of a linear programming problem in which the objective function is to be maximized subject to restrictions resistances 21, 22 are given conductance values representing c 1 , c 2  and resistance 23 is initially given zero conductance. If g 0  is the conductance of meter 19 and i 0  the current therein, it is shown that and potentiometers 15, 16 may be systematically adjusted from v 1  = v 2  = 0 to maximize the value of i 0  without allowing i 1 , i 2  to pass through zero, and if g 0  ” c 1 , c 2  the value of Z may be derived directly from meter 19, and those of v 1 , v 2  for optimal solution from meters 17, 18. Resistance 23 may be variable, and adjusted to zeroise meter 19, when the required value of Z is proportional to the conductance of 23 indicated on a calibrated dial. A simple meter may be switched to the positions of meters 19, 24, or 25. A more sophisticated form of the apparatus (Fig. 2) comprises the arrangement of Fig. 1 with resistance 23 replaced by connection over fixed resistor 27 from potentiometer 28 and voltmeter 37 across negative battery 14, and with amplifiers across meters 17, 18, energizing relays 33, 34 operating indicator lamp 35 when the polarity constraints of voltages v 1 , v 2  are infringed. Further columns and rows, with associated resistance elements, potentiometers, voltmeters and current meters, amplifiers and relays may be added. A removable link 36 is situated betweenpotentiometers 15, 16 and battery 13 and with potentiometer 28 zeroised enables the former potentiometers to be energized through a transistorized control element 32 excited from the transistorized amplifier 31 across meter 19, so that for positive potential between row 11 and earth, the energizing voltage of potentiometers 15, 16 retroactively tends to zero. As the setting of potentiometer 28 is increased, the reading of meter 37 indicating the required value of Z increases, and the controlled voltage on potentiometers 15, 16 also increases, and the combined effect of the current in resistances 21, 22 cancels the effect on amplifiers 31, 32 of the current in resistance 27, so that equilibrium is restored at equality of these currents, proportional approximately to Z. Initially, the resistance elements, 121 &c. and the settings of potentiometers 15, 16 &c. are predetermined, and the output of potentiometer 28 is increased, the voltage on conductor 38 rising accordingly until one of the constraints is infringed by meter 17 or 18 reaching zero, as indicated by lamp 35; after which adjustment of potentiometers 15 &c. restores compliance with the constraint. Potentiometer 28 is again increased, and the process is repeated until the number of potentiometers 15 &c. set otherwise than at zero equals the number of constraints satisfied, and no further adjustment is possible without infringing constraints. The optimal values of v 1 , v 2  &c. are derived from meters 24, 25 &c. and of Z from meter 37. Resistance elements 3, 6 are replaceable by networks similar to that of potentiometer 28, resistance 27, and meter 37, and the input of control 32 is connectable to amplifier 29 &c. Batteries are replaceable by A.C. generators with phase sensitive rectifiers associated with the relays and/or meters. The resistance elements and potentiometers are then replaceable by capacitance or inductance elements, and the columns may be connectable by variable mutual inductances. The apparatus is applicable to non-linear simultaneous equation solution and non-linear programming problems by replacing resistance elements 1 &c. by non-linear impedances, e.g. incandescent lamps or semi-conductor elements, so that the coupling terms a 1,1 , V 1  &c. become terms in squares and higher powers of v 1  &c. Potentiometers 15, 16 &c. are replaceable by switchable sources of discrete voltages from a tapped resistor or transformer, and the system may be arranged to solve minimization problems. The connections to the rows and to the columns may be arranged for transposition by plugs and sockets or switching. The impedance elements may be pre-set coaxial capacitors or resistances or may be resistive or capacitative elements having electrodes immersed in liquid or plastic material.