Patent GB-684989-A (GB)
Title: Improvements relating to electric circuit arrangements for performing mathematical operations
Inventor: LIEBMANN GERHARD
IPC: G06G7/46
Filed: 1950-07-25
Abstract: 684,989. Electric analogue calculating systems. ASSOCIATED ELECTRICAL INDUSTRIES, Ltd. May 29, 1951 [July 25, 1950], No. 18584/50. Class 37. An electrical calculating-device for solving heat transfer and temperature distribution problems in material under changing conditions as a function of time expressible by the equation where U denotes the temperature, t the time, # the Laplace operator, and k the material diffusivity (=#/sc, where # denotes the heat conductivity, c the specific heat, and s the density), operates by the iterative numerical solution of the equivalent difference equation to which the partial differential equation is reducible in the unidimensional case wherein U depends only on displacement x and time t, and in which #x, #t represent space and time intervals, U 0 , U 1 , U 2  represent temperature at positions x 0 , x 1 , X 2  separated by a space interval #x such that x 1 =x 0 +#x, x 2 =x 0 +2#x, &c., and the superfixed 0 and 1 of the temperature values indicate that they are taken at times t=t 0 - and t=t+#t. Equation (1) may be replaced by similar difference equations in the case of two or three dimensional systems, and the transformation is applicable to the differential equations of other thermal phenomena. The device comprises a resistance network provided with tapping points corresponding to positional points in the material, adjacent tapping points being interconnected by network resistances and connections being also provided between individual tapping- points and adjustable potential sources analogous to temperatures through tapping point resistances, the ratios of which to the network resistances are chosen to represent the thermal properties of the material at the corresponding position point, such that a temperature change of the material is represented by a potential applied suddenly to the corresponding point or points of the network and the resultant temperature distribution after a finite time interval is determined by the potential changes appearing at the tapping points, the process being iteratively repeated to determine the changes in temperature distribution after successive time intervals. Fig. 1 shows a network for the solution of the unidimensional problem of temperature distribution along a bar, exemplified by equation (2), wherein a chain of equal resistors Rx has tapping points X0X1 ... Xn corresponding to successive distances #x from a datum point x and resistances R0, R1 ... Rn having values given by are connected between the tapping points and the sliders of low impedance potentiometers P0, P1 ... Pn connected across the A.C voltage bus-bars U= 1, U=0. A probe Pr connectible to any of the tapping points is joined through a balance indicator M to the slider of a balancing potentiometer B across the busbars. If potentiometers P0, P1 ... Pn are adjusted so that the slider potentials T0, Tl ... Tn represent the temperature distribution at time t 0 , the resultant potentials U0, U1 ... Un at the network tapping points X0, X1 ... Xn represent the temperature distribution at time t 0 +#t, within the accuracy of the difference equation (2) above. Assuming the resistance chain represents, e.g. a bar of uniform thickness at initial temperature U=0, the potentiometers P1 ... Pn are initially zeroized and potentiometer P0 is rapidly adjusted to a voltage from the busbars on point X0 representing by analogy a sudden temperature rise of U 0 = 1 at the left end of the bar, the right end of which remains at zero temperature. Measurement of the potentials at points X0 ... Xn using the probe, balance indicator and potentiometer B gives potentials U 1 <SP>1</SP> ...Uln-i representing the temperature distribution after time t= #t. The potentiometers P1 ... Pn - 1 are now reset to impose corresponding potentials on the several tapping point resistances, and measurement of the new potentials U 1 <SP>2</SP> ... U n <SP>2</SP> at the tapping points indicates the temperature distribution at time 2#t, and the procedure is contained iteratively to indicate the successive temperature distributions at t=3#t, 4#t and finally the required result at t=m#t. The initial temperature distribution analogy potentials may be adjusted to any prescribed series of values and the temperature, analogy potential at x=0 may vary with time. The resistors R0 ... Rn may be variable in accordance with variations in thermal capacity or conductivity, or variations of the time or displacement intervals. If no intermediate values of the temperature distribution are required, duplicate sets of potentiometers P0 to Pn are provided, and are alternately connectible to the computing circuit by ganged change-over switches. The slider potentials corresponding to the initial temperature distribution are derived from the first set, and the slider potentials of the second set are adjusted to balance the various tapping point potentials after the first time interval. The changeover switches are operated to connect the second set in circuit, and the first set is adjusted against the tapping point potentials. The process is repeated for the prescribed number of time intervals after which the temperature distribution analogy potentials are measured as before using the probe indicator and balancing potentiometer. Fig. 2 shows such an arrangement applied to a single tapping point of a two-dimensional network of resistances Rx representing a plane or axially symmetrical heat transfer problem wherein the duplicated potentiometers P, P1 are adjustable by servomotors F, F1 and alternatively connected to the tapping point resistor Ri and the tapping point by ganged automatically driven changeover switches S1, S2 operating at time intervals #t. At the instant shown the slider potential of potentiometer P, corresponding to the tapping point potential at the end of the preceding time interval #t, is connected to the tapping point resistor, while the slider potential of potentiometer P1 is balanced by servomotor F1 against the prevailing tapping point potential. In the succeeding time interval the potentiometers will be interchanged, and the operation is continuous until arrested by the timing device after n operations when the temperature distribution analogy potential is measured, e.g. by an automatic measuring and recording probe scanning the several tapping-points. A single group of servomotors may be arranged to. operate successive groups of potentiometers in turn. The device is adaptable to the solution of the equation representing heat conduction with loss proportional to local temperature, e.g. along an uninsulated bar, by connecting each network tapping point to zero potential through an additional resistance given by where qi represents the local value of the loss constant q, which may vary with temperature or time, Fig. 3 (not shown). Ri<SP>1</SP> is adjustable manually or by a servo system after each time interval At, or may have a predetermined non- ohmic current/voltage characteristic, so that the generalized heat equation is soluble. The equation where F is a function of the space co-ordinates, and representing heat conduction with heat generation within the system is similarly soluble by connecting the appropriate tapping points to a voltage source pU 0  through a resistance given by Fig. 4 (not shown), in which Rill may be variable similarly to Ri<SP>1</SP> as described above. In a practical construction of a resistance network, Fig. 5 (not shown), parallel spring- tensioned potentiometer wires are stretched between the U=0 and U= 1 bus-bar, and have sliding spring finger tappings located in grooves in an insulated board, the interconnections being effected by jack switches.