Patent GB-884919-A (GB)
Title: Apparatus for evaluating algebraic functions of a complex variable
Inventor: MORGAN MERLE L
IPC: G06G3/00
Filed: 1958-03-25
Abstract: 884,919. Electric analogue-calculating; twopart couplings. ELECTRO-MEASUREMENTS Inc. March 25, 1958, No. 9450/58. Classes 37 and 38 (1). Electrical apparatus for evaluating quantitatively one of the real and imaginary components of an algebraic function including a complex parameter z and plural complex parameters a together with corresponding coefficients m, which function is an algebraic sum including terms each of which is a product of a coefficient m and a form function of the complex parameter z and a corresponding complex parameter a; such apparatus comprising first means establishing measurable physical quantities continuously representing the values of a pair of real and imaginary components of the complex parameter z, a plurality of second means, one for each term, and each establishing separate measurable physical quantities continuously representing the values of the pairs of components for a corresponding complex parameter a, third means operatively associated with the first and with the second means for establishing for each term a measurable physical quantity representing the value of the said component of the said form function of the complex parameter z and the corresponding complex parameter a, a plurality of fourth means, one for each term, and each establishing a separate measurable physical quantity representing the value of a corresponding coefficient m and each operatively associated with the corresponding second means for that term for establishing a measurable physical quantity representing the product of the coefficient m and said component of said form function of the complex parameter z and the corresponding complex parameter a for that term, and combining means operatively associated simultaneously with said pairs of second and fourth means for establishing a measurable physical quantity representing the value of said component of the algebraic sum of said terms. Theory.-A complex algebraic function wherein z is a complex variable, g is an arbitrary function thereof, a is a corresponding parameter and m a corresponding positive or negative real coefficient, and wherein f is a further function of z and a, is reducible to or to wherein F represents the whole of the stated function of z, K and a represent complex constant values, and n, m represent positive or negative real constants. In the invention individual measurable analogues of the real and imaginary components of a term representing the log of each factor without its exponent are developed, and multiplied by a coefficient representing the appropriate exponent; the resultant analogues representing similar components being combined in appropriate sign to yield a pair of measurable analogues representing the logarithmic real and imgainary components of the function. Fig. 1 shoes a co-ordinate system 20, 22 representing the real or the imaginary components of a function of a complex quantity w by the distribution of a voltage or other physical analogue over a field; the analogue at a point represented by each value of w being proportional either to the real or the imaginary component of log (z - w) or log z (1--) wherein z is an arbitrary complex w parameter. At a particular point w = a, the analogue represents real or imaginary com- ponents of (z-a) or (1z-/a), so that analogues at three points a 1 , a 2 , a 3  are multipliable at 24, 26, 28 by coefficients m 1 , m 2 , m 3  and the results are combined at summation unit 30 with analogues of a constant K from source 32 and of z<SP>n</SP> from source 34 and multiplier 36, so that the output at meter 38 calibrated in magnitude or phase represents in analogue form the equations (2) or (3) above. The field is limited by boundary 40 and by an inner circular region surrounding w = z at which the z  real component of log (z - w) or (1--) becomes w infinite and the imaginary component becomes indeterminate. The analogue of w may be a distribution of voltage over the field, or of physical displacement normal thereto and interpreted by transducer means, and in a preferred system the field co-ordinates are the respective real and imaginary components of log w and the parameter z is an arbitrary point of the field, wherein voltage distributions are set up such that the voltages in the distribution respectively represent the real and imaginary z components of log (z - w) or (1 --), so that at w any arbitrary point w = a, the voltage represents real and imaginary components of z log (z - a) or log (1 --). Each real component a voltage at points a 1 , a 2 , &c. is applied to an admittance proportional to arbitrary real factor m 1 , m 2 , &c. and the resultant currents are combined with the current in an admittance representing a real factor n due to an auxiliary voltage proportional to parameter z and a manually adjustable current proportional to the real component of the log of a constant K to represent the real component of log F. (Equations 3, 4 above). Similarly, the imaginary component voltages at corresponding points of the distribution and an auxiliary voltage representing the imaginary component of log z are applied to similar admittances and combined in a similar measuring circuit with a current representing log K to a total current representing the imaginary component of log F (the phase of F). The voltage distributions are established upon finite sheets of resistive material whose conducting and insulating boundaries conform to segments of contours of voltage and current respectively which contours represent the components of the functions set forth and, voltages proportional to corresponding comz ponents of log (z - w) or (1 - -) are applied w along the boundaries to produce the voltage distribution representing the appropriate component of the required quantity. The conducting boundaries of the finite sheet representing magnitude variables conform to contours of the real component of log (z - w) and insulating boundaries conform to contours of the imaginary components of that quantity, while a reference point # is chosen so that the magnitude of complex quantity # is << that of parameter z, so that voltage at a relative to that at # is   proportional to real component of log z-a/z;   the voltage at # referred to an ultimate reference point being made proportional to the real component of log z so that the voltage at a relative to such reference point is proportional to the real component of log (z - a). The voltages at points w = a 1 , a 2 , &c. are connected over admittances of values m 1 , m 2 , &c. while the voltage at point w = # is connected over an admittance representing exponent n, all to the ultimate reference point; the resultant currents being combined with a current representing log K to represent the real component of log F equivalent to the log of the magnitude of F. For evaluation of the imaginary component of log F there is provided a second sheet of uniformly resistive material on which the voltage at any point w = a relative to the voltage at any reference point w = # is proportional to the imaginary component of log z-a/z# and the conducting boundaries of the phase sheet conform to contours of the imaginary component of log (z - w) while its insulating boundaries conform to the contours of the real components thereof. Reference point # is selected so that the magnitude of # is << that of z, the voltage at a relative to # is proportional   to the imaginary component of log z-a/z, so that if the voltage at # relative to the ultimate reference point is proportional to the imaginary component of log z, the voltage at a relative to the ultimate reference point is proportional to the imaginary component of log (z - a). By connecting the voltages at points w = a 1,  a 2 , &c. and w = # to their respective admittances as before and combining the respective currents, the total current represents the imaginary component of log F and thus the phase of F. Where the function comprises factors (1-z/a)   the current in the sheet must flow from w = zero to w = z, and the voltage at any point w = a with respect to the voltage at reference point w = # is proportional to the real component of log and the conducting boundaries must conform to contours of the real component of log z (1 --) while the insulating boundaries conw form to contours of the imaginary component of the same. Reference point # is chosen so that the real magnitude of # is >> that of parameter z, the voltage at a relative to # is z proportional to the real component of log (1 - -) a and reference point # is connectible direct to the ultimate reference point, together with the current combining circuit. The voltage representing the real component of log z is derived from a voltage at a point S such that S << z and the voltage thereat relative to the ultimate reference point is proportional to the real   component of log (1 - z/S) or log (-z/S), and from this voltage is subtracted a voltage proportional to the real component of log (-1/S) to yield a voltage proportional to the real component of log z. This voltage together with those at points w = a 1 , a 2 , &c. are applied to the respective admittances representing n, m 1 , m 2  &c. and the resultant currents are combined to represent the real value of log K; the total representing the real component of log F. In this case, the phase sheet voltage distribution has a value at any point w = a relative to the reference voltage at w = # proportional to the imaginary component of log The phase sheet conducting boundaries conform to contours of the imaginary component z of log (1 --) and the insulating boundaries w conform to the contours of the real component of the same. As before, a voltage proportional to the imaginary component of log z is derived and applied together with the voltages at point w = a 1 , a 2 , &c. to admittances representing n, m 1 , m 2 , etc.; the resultant currents