June 1931 Radio-Craft
Wax nostalgic about and learn from the history of early electronics.
See articles from Radio-Craft,
published 1929 - 1953. All copyrights are hereby acknowledged.
Here is yet another treatise on the subject of reactance and
resistance. Considering that the date on this article is 1931,
it was probably amongst the first to publically discuss such
'newfangled' topics outside of a formal university setting.
The layman just was not accustomed to being bothered with such
esoteric concepts. After all, not many decades previous a person
might be burned at the stake for exercising such witchcraft
as speaking of 'imaginary' numbers as is required for a complete
analysis of alternating circuits. This article, however, does
not actually get into complex number, but future ones did.
The Whole Ohm Family - R, X and Z
A Simple Explanation of Reactance and Impedance
By Hal Wyman
There are several different kinds of ohms - as many readers
already know, and others have suspected from some of the theoretical
formulas which they have encountered in their studies as Service
Men, set builders and experimenters. That is to say, the ohm
is not merely the unit of resistance (R), which, under a constant
potential of one volt, allows one ampere to flow; but it is
also the unit of reactance (X) and of impedance (Z). In order
to explain the latter terms, let us first consider the fundamental
nature of an alternating current; and, in connection with the
latter, the common expression, a "sine-wave."
Fig. 1 - Left, a single "sine." Middle, a
flock of stiles, each corresponding to its respective anole.
Right, the conventional "sine wave" showing how it corresponds
to the circle, and how the current (or voltage) fluctuates during
the different portions of an A.C. cycle.
The Sine Wave
The sine is the distance between two sides of an angle; measured
in terms of the length of one side, on a perpendicular dropped
to the other side. See Fig. 1A, in which the line OA is the
radius of a circle. Its length is r; the length of the line
AB dropped from A perpendicularly on OB is y, then the sine
of the angle between OA and OB is y/r. It is customary, in calculations,
to take the length of the radius OA as 1; in which case, the
sine will always be represented by a fraction the greatest possible
value of which is 1.00.
For every possible angle, there is therefore a corresponding
ratio called its "sine." The angle may be carried beyond 90°,
around to 360°,. and it may be increased still further as
the radius continues to revolve around 0; but the numerical
values of the sine will repeat at every quarter turn though
alternately positive and negative. Similarly, in radio graphs,
where alternating voltage and current are represented by curves,
different polarities of voltage and directions of current flow
are indicated by the spaces above and below a zero line drawn
horizontally through the figure.
In Fig. 1B, we have a circle in which a radius is revolving
around the center 0, like the hour hand of a clock but in the
opposite direction here; the height y of its end A above the
zero line X-X, indicates the value of an alternating current.
If the maximum value of that current is taken as 1, when the
radius is pointing straight up, its value at any other moment
will be indicated by the sine of the angle through which the
radius has move
Every conductor (even a straight wire) carrying
all electric current creates a magnetic field, as indicated
at the right.
But, in drawing a graph like this, we make it difficult to
measure the movement of OA after it has completed the first
revolution; and so, to represent an alternating current, it
is customary to suppose that the center of our circle is moving
steadily along the line X-X (as in Fig. 1C. Then we call determine
the time during which the alternations have been taking place,
by measuring straight along X-X, which represents time. At the
same time, measuring parallel to the line Y-Y, we determine
current values which, in a true sine-wave, will be in exact
proportion to the sine of the time-angle, or clock-hand, of
Fig. 1B. This is called "plotting amplitude against time."
Inductance makes current "lag" behind voltage,
as at the left: capacity makes current lead the voltage, as
at the right.
When the clock hand of Fig. 1B passes 90° (which represents
the first quarter of an alternating-current wave) it begins
to come closer to the line X-X; and when it is right on the
line again, zero current is represented. As the moving hand
gets farther from this line, the current again increases; but
it is reversed, as compared with its former polarity. When it
gets to 270°, the current is at its negative maximum; and
when it is again 00deg;, having completed 360° of turn,
current is again zero, and the first cycle is over.
Graphic methods of computing impedance; left
upper, from resistance and inductance, right upper, resistance
and capacity. Below, the resultant of all three factors.
The same conditions are represented, but by a line which is
pulled out into a "sine wave," instead of by a circle, in Fig.
1C. If we are speaking of 60-cycle current, the time represented
by this curve is 1/60 of a second; if we are speaking of 600-kilo-cycle
current, the time represented is 1/600,000 of a second. But,
in either case, if we are dealing with pure sine-wave A.C. voltage,
the curve will be of the same shape.
The Effect of Inductance
For the moment we will leave our pretty pictures and attempt
an explanation of the term inductive reactance - pausing, of
course, for a brief definition of the word inductance. Suppose
that we have constructed a coil of wire wound around either
an air core or on a pile of iron laminations, and connected
as in Fig. 2. Magnetic lines of force are set up - our coil
has become an electromagnet. Now this inductive action will
not be purely external, but there will be also interaction between
the turns of wire within the coil itself. The strength of the
magnetic field is proportional to the number of turns and to
the current flowing. If the current is varied, the strength
of the magnetic field will also vary in direct proportion.
It is only when the current is changing that this secondary
effect, due to the common linkage of the turns, takes place.
While the current is changing, Lenz's law informs us, the induced
E.M.F. or voltage acts in such a way as to oppose the change
which is taking place. Mark the fact that the direction of the
current's flow (whether positive or negative) or its magnitude
have nothing whatever to do with this - it is the rate of change
in which we are interested.
Look again at the curve in Fig. 1C, and you will see that
the current's magnitude is continuously changing, except at
the instant in each half-cycle when the maximum positive or
negative value is reached.
Since the inductive effect is at maximum when the "rate of
change" is greatest, we may assume that the opposing E.M.F.,
due to the inductance of the circuit, is at its peak when the
current wave is passing through its zero value; and we can plot
a curve to this effect as in Fig. 3. The "peak" value of the
current is as shown by the curve, while the R.M.S. (root-mean-square)
or effective value is but 0.707 of the maximum (ordinary A.C.
meters read R.M.S. values.)
At any instant the slope of the current curve gives the rate
at which the current is changing; and it can be seen that the
slope is steepest a t the points where the current crosses the
zero line. We may plot the dotted "sinusoidal" line shown in
the Fig. 3 as representing an arbitrary relation between the
current and the "back" or "counter" E.M.F. in an inductance
coil of L henries.
Since the back E.M.F. is defined as being exerted in a direction
opposing the current change, we also have shown the back E.M.F.
as negative in value while the growth of current is in the positive
direction. The curve of this induced voltage (the back E.M.F.)
is seen from the figure to be just one quarter-cycle out of
step with the current; and we say that it is out of phase by
90 degrees, and that the current "lags the voltage" or that
the voltage "leads" by 90 degrees. This brings us up to the
problem of phase difference from which we will beat a hasty
retreat for the moment.
If the frequency in cycles is taken as f, the voltage
induced by the changing current is equal to 6.2832 times
f times the inductance in henries times the current in
This leads us back to Ohm's Law for direct current; and we
will find that, with the frequency set at a single value, the
equations are parallel to those of Ohm's law but with XL (the
inductive reactance - 6.28 f L) taking the place of
R; so that just as simply in the case of direct-current calculations:
(To be continued in July Radio-Craft)
Posted September 15, 2015