Origin and meaning of circular and hyperbolic functions in electrical engineering

Abstract

Sinusoidal currents in time, arising from impressed sinusoidal voltages in time, need no explanation; but circular and hyperbolic functions of unobvious circular and hyperbolic space angles arising apparently spontaneously in electrical circuits call for some explanation. Accordingly, the concept of an angle, adequate both for circular and hyperbolic functions, is first developed geometrically, pointing out that an angle is nothing but the percentage value (more precisely, percentage/100 or per-unit value) of an arc recognized as the vector increment of the radius vector. Naturally, the radius vector itself is used as the per-unit base, hence the name radians (meaning, radii) for this kind of measure numbers. The idea naturally leads to the compound-interest like calculation of the per-unit value of a gross change in the radius vector. The vector concept recognizes as angles those incremental arcs that are parallel to the radius vector just as well as those that are normal. The former are called hyperbolic angles, the latter circular. Thus an arc that represents a circular angle and one that represents a hyperbolic angle are j with respect to each other. Obviously, in the first case, only the direction of the radius vector changes; in the latter, only its magnitude. Applied to a physical variable, this concept of an angle means that a change in the variable expressed in per-unit 2 2 All such reckonings in this paper are to be understood as compound interest calculation. is the angular value of that change in radians. This concept of angles leads to logarithms and exponentials, yielding a simple interpretation of the logarithms of negative, imaginary and complex numbers so that, in its light, by inspection log (− 1) is recognized as πj, log j is recognized as ( π 2 )j , etc. The cosine of an angle θ (whether θ is circular, or hyperbolic, or mixed) is interpreted as the resultant of two variables, initially alike, but one having undergone the vector per-unit change (+ θ), and the other (− θ). A sine is interpreted as the resultant of two variables, initially equal and opposite, and one having undergone the vector per unit change (+ θ), the other (− θ). The origin of such pairing of vectors is found in reflection. In transmission lines, discontinuities like open circuits and short circuits act as perfect reflectors, and provide reflected variables which match the original variable in the indicated manner to yield with the original a resultant corresponding to the sine or cosine (or sinh or cosh) of the per-unit change involved.