Patent GB-669814-A (GB)
Title: Improvements in or relating to the electrical analysis of a physical system
IPC: G06G7/38
Filed: 1947-09-26
Abstract: 669,814. Electric analogue calculating systems. WESTINGHOUSE ELECTRIC INTERNATIONAL CO. Sept. 26, 1947 [Sept. 28, 1946], No. 26148/47. Class 37. In an electrical analogue computer for determining the response of a physical system, having at least one non-linear parameter, to the operation therein of a predetermined static or transient forcing function represented by an electrical waveform, the latter is applied to an electrical network simulating the physical system according to a consistent scheme of analogy, wherein the linear parameters of the physical system are represented by linear electrical quantities and the non-linear parameters by the combination of an input impedance chain associated with at least two electronic multipliers, the first multiplier having two inputs each connected across an element of the impedance chain, the second multiplier having two inputs connected respectively to the output of the first multiplier and across an element of the impedance chain; so that the resultant circuit represents at least three terms of a power series equivalent to the non-linear parameter. The behaviour of the various parameters of the physical system in response to the forcing function is determinable by connecting oscillographs or measuring instruments across corresponding elements of the analogy network and observing the readings or traces on excitation by the energizing waveform, which may be derived from a variable frequency signal generator, a magnetic tape recorder, or by synthesis from Fourier series components obtained by sequentially discharging plural capacitors through predetermined circuit networks and adding the several components. The behaviour of a non-linear system, e.g. a transiently excited circuit comprising an inductance in series with a non-linear resistor, is represented by the differential equation and Fig. 5 shows the analogy circuit in which a waveform e(α(f(t)) is applied to the inductance L(αA 1 ) in series with the networks M 1 , M 2 , M 3  representing a non-linear resistor whose value is a function of current. Postulating iα dx/dt, the equation reduces to Since a non-linear quantity may be represented to a first approximation by the first two or three terms of a power series, and since the value of a non-linear resistance is a function of the current therein, so that the non-linear resistor may be represented by the circuit of Fig. 4, wherein three electronic multipliers M 1 , M 2 , M 3  having gains k 1 , k 2 , k 3'  and a chain of series resistors R 1 , R 2 , R 3 , R 4 , R 5 , R 6 , R 7  carrying a current i are combined so that both inputs of M 1  are connected to R 2 , the output of M 1  is connected to R 3  and one input of M 2 , the other input of M 2  is connected to R 4 , the output of M 2  across R 5  is connected to one input of M 3 , the other input of M 3  to R 6 , and the output of M 3  to R 7'  The multipliers each comprise, Fig. 2 and Fig. 3 (not shown), an oscillator generating a carrier e 1  sin wt which is combined with input functions f 1 (t), f 2 (t) in successive balanced modulators to produce an output m 1 m 2 f 1 (t) f 2 (t)e 1  sin(w 1 t+#) applied to a linear demodulator together with a phase shifted unmodulated carrier e 1  sin (w 1 t+#) to produce a demodulated output f 1 (t)f 2 t representing the product of the two input functions. It is shown that the voltage E appearing across the resistor chain in Fig. 4 is given by By lumping constants and assuming the multiplier gains to be high so that the output impedances R 3 , R 5 , R 7  and the input impedances R 2 , R 4 , R 6  may be neglected, the equation reduces to which by suitable choice of constants represents equation (4) above so that the circuit of Fig. 5 represents by analogy the behaviour of a circuit comprising an inductance in series with a non-linear resistor when excited by a predetermined voltage waveform. By the standard electro-mechanical analogies set forth in the Specification, similar circuits may be set up representing mechanical systems embodying non-linear elements, e.g. a mass deflected by a transient force and slidably mounted with a non-linear coefficient of friction. In analogy networks for the solution of other problems the power series circuit equivalent to the non-linear parameter may embody a chain of inductive or capacitative impedances instead of resistors in the multiplier input and output circuits, whereby the voltages developed across each element will be e.g. L/d<SP>2</SP>q/dt<SP>2</SP> or Ldi/dt; q/c or l/c#idt; instead of R.dq/dt or Ri as in the above example. The behaviour of a compound non-linear system of two components in which a non-linear parameter of the first component reacts on the second component is represented by the simultaneous differential equations wherein the non-linear unilateral term in the equation of the second component represents the effect of the first component thereon. According to an electrical analogy set up in the defined manner, the co-efficient A 6 x 1  is represented by a non-linear impedance expressible as a power series and postulating i 1 α(dx 1 /dt) the non-linear unilateral term is similarly expressible as a voltage Thus the two simultaneous equations may be represented by the analogy circuit shown in Fig. 8, in which a predetermined forcing function f 1 (t) is represented by a voltage source e energizing a first loop circuit in which a current i(αdx 1 /dt)  flows through an inductance L 1 (αA 3 ) in series with three impedances Z 1  and a condenser C 1 (α 1/A4). The voltages across the several impedances Z 1  are respectively connected to an amplifier A, both inputs of a multiplier M 1 , and one input of a multiplier M 2  and the amplifier and multiplier outputs are connected in series with a second loop circuit comprising an inductance L 2 (αA 5 ), a capacitor C 2  (α 1/A7),and a voltage source e representing a predetermined forcing function; the output of multiplier M 1  energizing the second input of multiplier M 2 . If the gains of the amplifier and multipliers - are such that the outputs are large compared with the inputs, loop number 1 represents the equation (7) above and loop number 2 represents the equation (8) above since the voltages across the amplifier and multiplier outputs,add up to the first 3 terms of a power series representing the non-linear unilateral term. The electrical forcing functions may be derived from specific function generators or from preceding analogue computing circuits. The Specification refers to U.S.A. Specification 2,420,891, and to Guillemin's treatise on " The Mathematics of Circuit Analysis," Chapter VI, at pages 283, 284.