Patent GB-947233-A (GB)
Title: Improvements in equation solvers incorporating electric analogue computing means
IPC: G06G7/34
Filed: 1961-04-06
Abstract: 947,233. Electric analogue calculating. COMPAGNIE GENERALE DE TELEGRAPHIE SANS FIL, and SOC. MAROCAINE DE RECHERCHES,D'ETUDES ET DEVELOPPEMENTS SOMAREDE. April 6, 1961 [April 8, 1960], No. 12416/61. Headings G4A and G4G. Apparatus incorporating electric analogue computing means for solving systems of p non- linear equations in q unknowns of the type 1 2 comprise first means for providing q magnitudes respectively simulating the unknowns, having q respective control inputs and output means for q magnitudes; a feedback loop between the inputs and outputs for adjusting the respective magnitude values to the root values of the equations, such feedback loop comprising an error calculator having input means for collecting the q output magnitudes, means for computing the p equation values for the values of the unknowns generated by the first means; and p outputs providing p respective error signals; an electrical translating network having q outputs respectively designated by numbers j of values between 1 and q inputs respectively designated by numbers i of values between 1 and p, such network including transformer paths coupling each one of the q outputs to each one of the p inputs which paths are reversible of respective transfer coefficients #fi/#xj; the q outputs being respectively coupled to the q control inputs and the p inputs being respectively coupled to the p outputs of the error calculator. Fig. 1 shows an electric analogue computer for solving the generalised equations (3) (4) in which having the specific form (5) (6) wherein are substituted to reduce to (7) (8) for insertion into the computer. Servomotors G 1 , G 2  operate in response to control voltages U 1 , U 2  to rotate output shafts at speeds proportional to such voltages in rebalancing senses; the output shaft displacements representing X 1  and X 2  operating analogue error calculator DC to produce voltages V 1  and V 2  at outputs SC 1 , SC 2  given by (9) (10) where V 0  is a reference voltage; and also rotational outputs of shafts SC 1 , SC 12 , SC 21 , SC 22  representing (11) (12) (13) (14) derived from equations (7) (8) above. Voltages V 1 , V 2  respectively energize in series the terminals e, e<SP>1</SP> of transformers T in quadripolar networks 11 and 12; 21 and 22 respectively to excite sliders M of autotransformers T<SP>1</SP> centre tapped to earth. The sliders in quadripoles 11, 12, 21, 22 are respectively driven by shafts SC 11 , SC 21 , SC 12 , SC 22  to develop summation voltages from 11, 21; 12, 22 given generally by (15) where N ij  is the quadripolar transfer characteristic given by Nij=#fi/#xj (16) At equilibrium, equations (7), (8) are satisfied for values of X 1  = X 10  and X 2  = X 20  when U 1  = U 2  = 0 and it is shown that if the output shafts of G 1  and G 2  depart from the balance position to provide magnitudes X 1  X 10  =dx 1 ; x 2  = x 20  + dx 2  where dx 1 , dx 2  are small, the control voltages U 1 , U 2  take the values (17) (18) and the control voltages bear a constant ratio to the deflections of the output shafts from the equilibrium position, so that the system is stable and the shaft displacement at equilibrium represents roots x 1  and x 2 . Each quadripolar network may generally be reversible with q outputs and p inputs where q is the number of unknowns and p the number of equations to be solved, with the q outputs connected to the q inputs of the servo systems and the p inputs connected to the p outputs of the error calculator; arranged so that for unity voltage applied to any one of the q outputs a voltage representing #fi/#xj appears at one of the p inputs due to the built in reversible variable transfer characteristic #fi/#xj. It is shown with reference to Fig. 2 (not shown) that losses inherent in the quadripolar networks may be represented by resistance r in series with terminals e, e<SP>1</SP> of transformer T and resistance R shunted across transformer T<SP>1</SP>, and that the system operates with accuracy provided that # = 1/1-# # 1 where # = r/#N<SP>2</SP> for the quadripolar network. In a modification (Fig. 3 not shown) the inputs and outputs of the quadripolar networks are insulated from earth. Each quadripolar network (Fig. 4) may comprise a unity ratio transformer T supplying primary terminals e, e<SP>1</SP> and energized through series fixed capacitors C c  and C<SP>1</SP> c  (= 2C) and shunt inductance L s (=L) from a network comprising shunt inductances Lp(= L) and LM(= L/2)interconnected by variable capacitors CA = C<SP>1</SP> A  = C + d and C B  = C<SP>1</SP> B  = C - d where d is variable between O and C in response to a shaft B rotated by an output SCij of the analogue error calculator DC with # o <SP>2</SP>LC = - 1 and transfer coefficient =d/c variable from -1 to +1 with operating frequency # 0 . The parameter p is given by (C<SP>2</SP>/L<SP>2</SP> + 1)1/Q<SP>2</SP> where Q represents the quality factor of the inductance coils. Specification 703,178 is referred to. In a modification (Fig. 5) a circuit similar to Fig. 1 for solving two simultaneous nonlinear equations operates in response to analogue currents representing U 1 , U 2  and SC 1 , SC 2  with transformers T in the quadripolar networks replaced by parallel equal inductances L, L<SP>1</SP> coupled by capacitance C; the inputs of quadripoles 11,12 and 21, 22 being respectively energized in parallel and L, C being given by # 0  <SP>2</SP>LC = - 1 where # 0  is the operating frequency. It is shown with reference to Fig. 6 (not shown) that the inherent network losses in the quadripoles may be represented by admittances a shunted across transformer T<SP>1</SP> and b shunted across inductance L 1 , and that the system operates accurately provided In a modification (Fig. 7 not shown) the inputs and outputs of the quadripoles are solated from earth. The adjustable quadripolar network of Fig. 4 may be employed (Fig. 8 not shown). In further modifications (Figs. 9, 10 not shown) a system of equations (19) (20) (21) is solved by extensions of the computers shown in Figs. 1 and 5 wherein the error magnitudes are respectively represented by voltages and current intensities; servosystems G 1 , G 2  representing the roots x 1 , x 2  as before while the error calculator DC operated thereby produces three electrical outputs SC 1 , SC 2 , SC 3  and six rotational outputs SC 11 , SC 12 , SC 21 , SC 22 , SC 31 , SC 32  all fed to adjust the individual circuits of the quadripolar network DR. A mathematical analysis is given. Other modifications (Figs. 11, 12 not shown) comprise extensions of computers shown in Figs. 1, 5 to solve the system of equations (22) (23) there being provided three servosystems G 1 , G 2 , G 3  representing the roots G 1 , G 2 , G 3 , two electrical outputs SC 1 , SC 2  from the error calculator driven by the servosystems are fed to the quadripolar network DR, and six rotational outputs SCij are provided to adjust the individual quadripoles of the network DR. It is shown that the solutions are infinite in number with one degree of freedom and define the curve in space of the co-ordinates x 1 , x 2 , x 3 . The servomotor control voltages U 1 , U 2 , U 3  are zero when the servomotors produce values X 10 , X 20 , X 30  corresponding to a point on the curve. Figs. 13, 14 show systems corresponding to those of Figs. 1, 5 respectively, and adapted for solving p equation in q unknowns (24) (25) wherein q servomotors G 1  to G q  introduce mechanical rotations SR q  representing the roots of the equations into error calculator DC, which produces p electrical outputs SC i  representing the values of the several equations supplied to the quadripolar network DR; the individual quadripoles of which are respectively adjustable by q mechanical outputs SC ij  from the error calculator. Switches 10 l  . . . 10 p  and 20 l  ... 20 p  operate to disable the appropriate connections SC l  and SR q  when the numbers of unknowns and equations are respectively less than q and p. In Fig. 13 the quadripoles are of the type shown in Fig. 1 and in Fig. 14 are those of Fig. 5, and it is shown that for p = q the control voltages SC i  are zero when servosystems G 1  to Gq represent the roots by their rotations. Conditions for control voltage zero when p > q or p < q are also given, and in the latter case the servosystems represent one only of an infinity of solutions of the equations. Fig. 15 shows a modification for solving p equations in q unknowns wherein the quadripolar network DR of the kind described above produces analogue information signals U 1  ... U j  . . . Uq which are converted by devices TAD 1  . . . TAD j . . . TADq to corresponding digital signals which are applied to storage devices CP x1  . . . CP xj  ... CP xq  already containing digits x 1  . . . x j  . . . x q  operated so that the digital information signals U 1  ... U j  . . . U q  are added to the digits x 1  . . . x i  . . . x q  already stored and fed on as new digits x i ... x j  . . . x q  to a conventional digital computer DC deriving digits f 1  . . . f 1  . . . f p  representing the values of the several equations, and corresponding digits representing which are respectively stored in memory circuits feeding digital to analogue converters TDA 1 . . . TDA 1  . . . TDA p  and TDA 11  . . . TDA ij  . . . TDA pq  producing analogue signals f 1  ... f i  ... f p  and #f 1 /#X 1  ... #f i /#X j  ... #f p /#X q  for application as respective equation values and transfer coefficients to quadripolar network DR. Initially in response to a clock signal, CP x1  to CP x9  deliver digital data x 10  to X qo  to calculator DC providing corresponding functional data f 1  to f p  stores in Mf 1  to Mf p , and data The stores are clock controlled to feed out to converters TDA 1  to TDA p  and TDA 11  to TDA ij  providing analogue voltages f i  and mechanical rotations for the quadripoles of the networks DR as