Application of Heaviside's Resistance Operators to the Theory of the Air-core Transformer and Coupled Circuits in General

Abstract

In circuits possessing constant inductance, resistance and capacity the differential equations for the currents and the voltages are linear with constant co-efficients, and may therefore be solved by aid of the known simple properties of symbolic operators. The symbolic operator method proves to be very compendious in problems concerning the determination of the primary and secondary currents and voltages of transformers whenever the applied E.M.F. can be expressed as an exponential function of the time, or when it consists of a sudden application of a constant or an exponential function, and also when it is impulsive. In the Paper the method is first applied to a pair of indirectly coupled circuits - i.e., circuits that are insulated from each other. The cases of pure magnetic coupling, pure electric coupling and of mixed magnetic and electric coupling are treated in turn and shown to be operationally all of the same form. Then general operational equations are obtained for direct coupling and for mixed direct and indirect coupling - this latter, of course, includes all types of autotransformers, both high and low frequency. Here again the operational equations are of the same form as in the case of magnetic coupling. The equations readily show that if the primary applied voltage be a pure sine or cosine alternating E.M.F. the well-known circle diagram for determining the instantaneous magnitudes and phases of the primary and secondary currents in an ordinary transformer is applicable to the most general type of transformer, and is capable of certain extensions. It is then shown that a similar circle diagram can be used for obtaining the primary and secondary currents in any general transformer under the application of the quasi-steady conditions of a slowly-damped alternating voltage. A further new theorem arises here, which affirms that a similar graphical method of computation can be used for deducing the voltages at the terminals of the primary and secondary condensers in a high-frequency transformer under steady or quasi-steady alternating current. In the second section of the Paper it is shown how to calculate accurately the free periods and the damping factors of a coupled system with two degrees of freedom, and approximate formulae are also deduced. The third section passes to the investigation of the currents and voltages produced in the circuits by sudden application or removal of a constant, a harmonic, or a damped harmonic E.M.F., and of those produced by the application of electromotive impulses. For this purpose formulae expressing the effect of the various types of symbolic operator occuring in the Paper, on impulsive E.M.F.s and on sudden applications, are deduced and tabulated. Using these tables of integrals it is very easy to write down quickly the solutions appropriate to any given transformer under the action of any of the types of E.M.F. specified. These tables include all the cases likely to arise in wireless telegraphy and in ordinary transformers. As an example of the application of the methods the problem of a primary spark-circuit and its antenna is worked out under resonance conditions, and for various couplings and time graphs are made of the primary and secondary currents and voltages. A further example is the solution for the incidence of a natural electric wave (a stray) on a receiving antenna and its resonant secondary.