Equivalent Circuits for Oscillating Systems and the Riemann-Christoffel Curvature Tensor

Many electromechanical systems that are being studied in connection with the war effort have so many degrees of freedom that it is totally impractical to analyze their performance by direct calculation. It is, therefore, increasingly important to develop equivalent circuits for these multiple oscillating systems, that can be put on the a-c calculating board and so solved by direct reading of instruments. In this paper it is shown that a necessary (though not sufficient) condition for the existence of a physical model, corresponding to a given set of equations, is that the set should be a tensor equation. This follows from the fact that only quantities that are tensors can be measured by instruments. The conclusion is therefore reached that only equations that are in tensor form can be set up on the a-c calculating board. The principle is illustrated by setting up equivalent circuits with the aid of tensorial hunting equations for the determination of the steady-state, hunting, and self-excitation performance of two interconnected instrument-Selsyns, of two salient-pole synchronous machines, and of a capacitor-compensated transmission line connected to a salient-pole synchronous machine. A companion paper, ``Self-Excited Oscillations of Capacitor-Compensated Long-Distance Transmission Systems´´¹² contains the results of an extended study made on the a-c network analyzer with the aid of one of the equivalent circuits developed here.