IEEE Power & Energy Magazine - January/February 2020 - 81

In polar coordinates, a quantity at any time is repre-

textbook Theory and Calculation of Alternating Current Phe-

sented as a length and a time (with time being a repre-

nomena (1908), where a whole chapter (VI, p. 42) is devoted

sentation of angle for a periodic function.) It was obvious

to comparing the polar diagram and the crank diagram. Stein-

to Steinmetz that any periodic function could be repre-

metz knew about our sort of phasor diagrams, and he thought

sented in polar coordinates, and this is what he taught.

them impractical. In his vector power discussion, he wrote:

In Figure S2, a movable radius vector rotates with time,

This limitation to sine waves appears to me as one

starting from the abscissa at t = 0. (Approximately 100 years

of the main reasons, why a standardization of the

ago, there was much discussion on which way it should

crank diagram could not fulfil the purpose of secur-

rotate. Anticlockwise was settled on.) The moving radius

ing uniformity of notation, by eliminating the polar

vector traces out the circle in Figure S3.

diagram, since the latter would still have to be used

If the amplitude varied as a sine wave, a half cycle of

when dealing with distorted waves, as for instance
when working up oscillograph records, etc.

the waveform could be represented by a circle. The direction of the diameter from the origin gives the phase of

Of course, deep down, we know that the waves we deal with

the sine wave. The phase thus determined would require

are distorted. These days, we tend to overlook that truth.

that the sign of the phase offset term in (1) be negative.

For Further Reading

The other half of the sine wave would be drawn right on

C. P. Steinmetz, "Complex quantities and their use in elec-

top of the first half. If the wave were known to be a sinu-

trical engineering," presented at the Int. Electrical Congress,

soid, the circle could be omitted and only the diameter

Chicago, IL, 1893.

drawn. The resemblance to our diagrams is then strong-
but superficial.
The phasor diagram (as we know it today, centered on
the origin) is not the same. Our method, the crank diagram
to Steinmetz, is not mentioned until the fourth edition of his

The concept of the rotating phasor arises from the time dependence of the complex exponential
which characterizes AC voltages
and currents. Let us consider the
following phasor representation
for an AC voltage
V (j~, t) = Vs e j (~t + z) .
The concept of the rotating phasor does
not, in fact, "arise from the time dependence of the complex exponential" of anything; it arises from a book written more
than 100 years earlier by Lord Kelvin!
Has the line phasor gone away entirely? Perhaps. In a 1997 book, Electrical Engineering, Mishra describes the
location of some points on the complex
plane, writing, "If we join these points to
the origin, with arrow-heads directed toward the points, we obtain line segments
i.e., OA , OB , and so on which are called
phasors." I have seen nothing more recent referring to the phasor specifically
as a line. Some authors-perhaps most-
separate the argument of the exponent

A. E. Kennelly, "Vector power in alternating-current circuits," presented at the 27th Annu. Conv. Amer. Institute Elect.
Engineers, Jefferson, NH, 1910.
C. P. Steinmetz, Theory and Calculation of Alternating
Current Phenomena. New York: McGraw-Hill, 1908.

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IEEE Power & Energy Magazine - January/February 2020

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