Steinmetz Preferred Polar Coordinates A sinusoid, such as is given in (1), is a periodic function. The this was a paper written to oppose the current phasor method, great Charles Proteus Steinmetz understood that periodic which had existed since at least 1867. Steinmetz opened the pa- functions were well represented in polar coordinates. He per with the words, "In the following, I shall outline a method drew diagrams with axes and circles on them and called of calculating alternate current phenomena, which, I believe, them vector diagrams. It is often written that Steinmetz differs from former methods..." He then went on to discuss introduced the vector/phasor method to power systems. the method of radius vectors "known to every mechanical en- But in fact, Steinmetz's diagrams were very different from gineer from the Zeuner diagram of valve motions of the steam ours. Steinmetz did not regard our methods very highly! engine..." and bemoaned the fact that "the polar diagram has Our modern method of representing vectors (phasors) been utterly neglected." In his discussion of a 1910 work by Ken- is discussed in Steinmetz's paper "Complex Quantities and nelly, Steinmetz again described our current method as having Their Use in Electrical Engineering," read at the Interna- arisen separately from his paper, and he referred to the phasor tional Electrical Congress in Chicago Illinois in 1893. Here is diagram as we know it as the crank diagram, ostensibly because what he said after extolling the virtues of the polar method, it reminded him of the crank of a reciprocating engine. "with time as amplitude, and the instantaneous values of Steinmetz favored the use of the polar notation. We will find it unfamiliar, so let us illustrate it by means of a sine-wave the function as radius vector...": In its place diagrams have been proposed, where of current, as in Figure S1. (The figures are based on figures revolving lines represent the instantaneous values from Steinmetz's discussion of the Kennelly paper in 1910, and by their projections upon a fixed line, etc., which they have been renumbered for inclusion in this article.) diagrams evidently are not able to give as plain and He explains as follows: intelligible a conception of the variation of instan- We thus plot the current wave in polar coordinates: taneous values, as a curve with the instantaneous the angle, { in [Figure S1], is the abscissa of the rect- values as radii, and the time as angle. It is easy to angular coordinate representation ... that is, the understand then, that graphical calculations of alter- time t; the radius i is the ordinate, the current, volt- nate current phenomena have found almost no en- age, etc. This gives us as the polar curve, that is, the trance yet into the engineering practice. representation in polar coordinates [Figures S2 and We recognize the reference to "projections on a fixed line" as S3], of the [original] sine wave... . describing modern diagrams. This 1893 paper is sometimes cited The circle (Figure S4) thus is the general representation of as being the one that gave the world the phasor method. In fact, the sine wave in polar coordinates. And there is the surprise: the circle is not centered on the origin; it passes i through the origin. θ i1 e Tim i1 ϕ O t1 t A O figure S3. A complete current wave in polar coordinates. figure S1. The sine wave used by Steinmetz. Ti e m i1 e Tim t1 or ϕ O θ A figure S2. A snapshot of a current wave in polar coordinates. 80 ieee power & energy magazine O A figure S4. A sine wave in polar coordinates. january/february 2020

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