Definition and properties of a Eulerian 3-terminal gyrator

Abstract

The Eulerian 3-terminal gyrator is a hypothetical element whose novel features relate to systems dominated by dynamic forces. In mechanical linkages, turbomachines, flow networks and many other systems forces are proportional to the square of velocities. This square-law is fundamental in the Eulerian 3-terminal gyrator, unlike the linearity of the conventional gyrator. A gyrator embodies non-reciprocity and, without power loss, exchanges pressure-like and flow-like variables. The name derives from the gyroscope, for which torque and precession rate are linearly related if the gyro speed is constant, but this speed would have to vary to produce a square law relationship. Such so-called ‘internal modulation’ has been viewed askance in the definition of an ideal element. However, despite potential mathematical complexity, a neat definition follows from a coordinate transformation and mapping relating to 3-terminal elements. The resulting characteristics are well-behaved functions. Formulae are derived for the small-signal Z - and Y -parameters and these enable the non-reciprocity to be demonstrated. The large-signal characteristics are compared with those of a fluidic reverse flow diverter applied to process fluid handling. In general, the Eulerian gyrator is proposed as a natural prime element in analytical modelling because it builds in the modulation that must be applied to a linear gyrator in representing systems beyond the confines of electrical networks.