The Application of the Laplace Transformation and a Mechanical Harmonic Synthesizer in the Analysis of Electric Circuits

Abstract

The purpose of this research was to demonstrate the applicability of a mechanical harmonic synthesizer to the solution of the operational equation related to the analysis of electric circuits. Specifically, the machine has been used to determine the roots of the polynomial in s (the derivative operator) found in the denominator of the equation ik = Ek(s)/D(s). In order to achieve this purpose, the mesh currents of several electric circuits, in which the numerical values of the circuit elements were known, were computed by use of the Laplace transformation method in conjunction with the mechanical harmonic synthesizer. In designing a circuit to demonstrate a specified method, it was observed that the character of the roots of the polynomials could be predicted by use of the facts which follow: 1. For every mesh that consists of R-C elements or R-L elements there is one real root. 2. For every mesh that consists of R-L-C elements there are: a. At least one real root when R/2>(L/C)½; b. Two real roots when R/2 = (L/C)½; c. Two conjugate complex roots when R/2<(L/C)½ where R, L, and C are the effective values of the resistance, inductance, and capacitance of the elements in the mesh.