IEEE Power & Energy Magazine - January/February 2020 - 80

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

through the origin.

Tim

i1
ϕ
O

t
A

figure S3. A complete current wave in polar coordinates.
figure S1. The sine wave used by Steinmetz.

Ti
e
m

Tim
t1 or ϕ
O

θ
A

figure S2. A snapshot of a current wave in polar coordinates.

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figure S4. A sine wave in polar coordinates.

january/february 2020

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