Abstract—Generally applicable accurate diode voltage selectors are described as well as means for the reduction of static errors in cascade multipliers and in multiplier input converters. An analysis of static errors is given.

A new principle and circuit for the generation of the product of two variables is described in which the required result is a linear combination of readily produced secondary variables. The product of two variables can be expanded in a series in which each term is derived from the preceding one and from an auxiliary variable by linear operations and by maximum and minimum selection. Two ``converted'' variables are associated with each term of the expansion. The product is equal to a linear combination of the converted variables plus the product of the last two converted variables. The range of variation of successive converted signals decreases by half so that the series converges by factors of ¼ per term. The exrpansion stands in direct correspondence with electronic multipliers which use linear elements and diode selection circuits. Only two sign-changers are required for four-quadrant operation. When used in conjunction with an ancillary four-quadrant multiplier, the cascade multiplier increases its accuracy by factors of 4, 16, 64,..., if 1, 2, 3, ..., stages are used. Alternatively, a sufficient number of stages provides a complete multiplier even without the use of an ancillary multiplier. Accuracies of the order of ±0.1 per cent are readily obtained and speed of response is high.

Mutual inductance is used to reduce the effect of inductance and thus increase speed. If wires are close together when current is switched from one to another, there is very little change in magnetic field, and consequently, very little energy is necessary to do the switching. Therefore, the effective inductance is small. The required close coupling is obtained by depositing one wire of the pair over the other instead of side by side, as is usually done. This permits improvements in electrical time constant, (irreversible) thermal repetition rate limit, and reliability. There are cryotron configurations suitable for use with inductively coupled lines.

When diode selection circuits are fed with signals that vary linearly with an input variable, they produce as output a convex or concave function consisting of one linear section per diode. In the present circuits, by combining each diode with a suitable series ``interpolating'' resistor, and feeding the output connection with constant current, it is possible to produce 2n-1 linear sections with n diodes. The use of voltage dependent interpolating resistors replaces the linear sections by nonlinear curves making possible the generation of smooth nonlinear functions. The function generator is stable as compared with similar nonlinear circuits because its transfer characteristic is independent of the nonlinearity of the resistors in a first approximation. Accuracies of the order 0.5 per cent of full scale output are readily obtainable even in the simplest circuit configurations. 1° phase shift occurs at frequencies of the order of 10 kc for nonlinear interpolation and up to several hundred kc for linear interpolation. The representative example of a squarer is described in detail; an error analysis is given and it is shown how the circuit can be compensated for a constant linear load.

Matrix formulation permits the compact analysis of a very general computing scheme based on operational amplifiers. The computer solves an equation in which a computing matrix and an error matrix can be distinguished. Programming a differential equation consists of writing it in an appropriate matrix form. The computing setup is in immediate mutual correspondence with the computing matrix. The error matrix can be written down by inspection.

A frequency error analysis of computing elements is presented which leads to a definition of their dynamic accuracy. The concept of a computing transfer function is introduced for this purpose, permitting the evaluation of an effective bandwidth, the latter being connected with the variance of the output for wideband inputs. Limited bandwidth is considered as equivalent to finite resolution and thus to an additional effective error. Single frequency errors are dealt with separately and are shown to be of minor importance. Suitable optimalization of dynamic accuracy yields parameters of design and performance such as optimum computing time and required base amplifier gain. The theory is applied to integrators and adders with base amplifiers of direct and of capacitive coupling.