January 1939 Electronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from Electronics,
published 1930  1988. All copyrights hereby acknowledged.

You might recognize the name
Phillip H. Smith.
He was an engineer at the Radio Development Department of Bell Telephone Laboratories.
This January 1939 issue of Electronics magazine might be the first publication
of what is now the most recognized form of his famous
Smith Chart.
Here, Mr. Smith describes the motivation behind his new chart and the process
used to create it. Being a humble man (a presumption), he titled his invention the
"Transmission Line Calculator," rather than naming it after himself. The Smith Chart
enjoyed wide acceptance fairly quickly by those who understood how and why it worked.
Analog Devices now owns the copyright to the Smith Chart; however, the company must
not guard their ownership of its likeness too jealously since many thousands  probably
hundreds of thousand or millions  of instances appear everywhere in print, online,
and in software. Maybe some rambunctious lawyer will someday convince Analog Devices
to collect royalties. If you hear of it in the works, be sure to buy
ADI stock. The January 1944
issue of Electronics magazine carried a followon article by Mr. Smith
entitled, "An Improved Transmission Line Calculator."
This might be the first publication of what is now the most recognized form
of his famous Smith Chart.
Fig. 1  Rectangular coordinate system forming the basis of
the transmission line calculator. Resistance components plotted along abscissa,
reactive components along the ordinates.
Fig. 2  Rectangular coordinate system similar to Fig. 1,
but with dotted curves representing the locus of points at specified distances from
the point of maximum impedance of the line.
By Phillip H. Smith
Radio Development Department Bell Telephone Laboratories
Transmission line problems are greatly simplified if the line is terminated by
an impedance equal to its characteristic impedance. Standing waves of current and
voltage are then eliminated, and if losses are neglected, the input impedance, current,
and voltage at all points along the line are constant. Computations for the current
and voltage under these conditions are based simply on Ohm's law. In communication
systems this special condition is usually considered most conducive to efficient
troublefree operation.
There has long been a need for a simple means, without recourse to lengthy computations,
for evaluating the impedance, current, and voltage at any chosen point along radiofrequency
transmission lines in terms of specific values of the several transmission line
parameters. This has led to the development of a special radiofrequency transmissionline
calculator for solving many ordinary transmissionline problems.
There are four factors that generally enter the solution of problems involving
the changing input impedance along a line. These are the characteristic impedance
of the line, the load impedance, the length of the line, and the input impedance.
If any three of these are known, the fourth may be found from the relationship:
Where L is the length of the line in wavelengths, Z_{0} is its
characteristic impedance, which for lowloss radiofrequency lines is essentially
a pure resistance, Z_{r} is the load impedance, and Z_{s} is the
input impedance.
The independent variable Z_{0} will be a constant for any particular
line, and consequently may conveniently be combined with Z_{s} and Z_{r}
in the form of a ratio. With this transformation, the equation becomes:
For any one value of Z_{r}/Z_{0} substituted in this equation
a series of values for Z_{s}/Z_{0} can be plotted for various values
of L from zero up to onehalf wavelength. Because L appears in the equation as an
arc tangent, the curve will repeat itself for every half wavelength. If such a curve
is plotted on a rectangular coordinate system with the resistance components along
the abscissa axis and the reactance components along the ordinate axis, it will
be found to be a circle with its center on the resistance axis. For other values
of Z_{r}/Z_{0} other curves could be drawn, and all would likewise
be found to be circles with their centers on the resistance axis, but they would
not be concentric. Such a set of curves is shown in Fig. 1.
Each of these curves, however, represents more than a single value of Z_{r}/Z_{0}.
At the load end of the line,. for example, where L = 0, Z_{s} will be equal
to Z_{r}, which determines one point on the curve. Obviously every other
point on the curve corresponds to another value of Z_{r}, when L is taken
as zero at that point. This other value of Z_{r}, when substituted in the
above equation, will give the same curve but with a different position for the point
of L = 0. These curves, therefore, cannot be completely designated by a single value
of Z_{r}/Z_{0}. It will be noted, however, that each curve gives
a minimum and maximum value of Z_{s}/Z_{0} which are the two points
where the curve crosses the resistance axis. The ratio of the minimum to the maximum
value of Z_{s}/Z_{s}, which is the same as the ratio of minimum
to maximum Z_{s}, is different for each curve, so that each may be designated
by the ratio of minimum to maximum input impedance that is peculiar to it. Each
circle thus represents the input impedance over a half wavelength section for a
family of transmission lines of various values of characteristic and load impedances,
but alike in having the same ratio of minimum to maximum impedance. The ratios of
minimum to maximum current or voltage will also be the same for any transmission
line represented by a single curve, and may be designated by ρ, and these values
of ρ are marked on the curves.
Fig. 3  Circular form of transmission line calculator. The arm,
shown below, is to be pivoted at the center of the circular chart. For convenience,
a transparent slide may be slipped on the rotating arm.
The points of minimum and maximum impedance on each curve are onequarter wavelength
apart, and all other points on the curves may be indicated as fractional wavelengths
from the points of minimum or maximum impedance, as marked on the curves. If the
points on the various curves at the same distance from the points of maximum impedance
are connected by a line, the plot will appear as in Fig. 2. These distance curves
are also circles; their centers are on the reactance axis and they intersect the
original family of circles at right angles. Each of the these curves is marked with
a value for L which represents the distance in wavelengths to the point of maximum
impedance. Such a set of curves could be used to solve various transmission line
problems. Assume, for example, that the resistance component of the load impedance
was 2.7Z_{0} and the reactance component of the load impedance was 0.9Z_{0}.
This value of impedance is found to lie on the circle marked ρ = 0.333. The intersection
at this point of the circle marked L= 0.02 simply indicates the distance to an arbitrary
reference point where the impedance is maximum. This point on the curve actually
represents the conditions at the load end of the linewhere L. 0. The impedance
at any other point, for example .08 wavelengths toward the generator, may be found
by following clockwise around the circle marked ρ = 0.333 the required distance
to the intersecting circle where L (0.02 + 0.08), or 0.10. At this point the impedance
will be found to be 0.8Z_{0}  j 1.0Z_{0}.
This information may be arranged in a much more convenient form, however, by
a transformation of coordinates that converts the rectangular mesh of resistance
and reactance coordinates into coordinates comprising two families of circles intersecting
each other at right angles. If the curves of constant ρ, and constant L were plotted
on these transformed coordinates, the ρ curves would be found to be concentric circles
and the L curves would be equally spaced straight lines radiating from the center
of the ρ circles. Instead of actually plotting the curves, however, an arm may be
pivoted at the center of the ρ circles, and by providing a slide on this arm, any
of the ρ circles may be described merely by rotating the arm with the slide fixed
in the proper position. By the addition of an adjustable distance or L scale around
the periphery, the impedance at any point along the line may be readily determined.
A calculator of this type has been constructed to facilitate the solution of
transmission line problems, and is shown in Fig. 3. The resistance and reactance
coordinates are marked as a ratio to Z_{0}; thus a marking of 1.6 on a resistance
circle means a resistance of 1.6 times Z_{0}. The arm is graduated with
a scale for p, and so the slide can be set to any desired value. The distance scale,
which rotates around the periphery of the diagram, is marked from 0 to 0.5 wavelength
by two scales reading in opposite direction, so that distance can be read in either
direction from any initial starting point on the coordinate system.
In using this calculator the length of the line and its characteristic impedance,
Z_{0}, will, for example, be known. If in addition both the resistance and
reactance components of the impedance 30 at any point of the line are known, the
values of the impedance at all other points, including the input impedance, and
the location of the points of minimum and maximum impedance, current, or voltage,
are readily found.
Assume for example that the load impedance is known. The corresponding impedance
point would be located on the coordinates of the calculator, and the arm and slide
would be moved until the center line of the arm and the cross line of the slide
intersected over this point. The ratio of minimum to maximum voltage, or current
could then be read directly from the scales on the arm. The zero point of the outer
distance scale would then be set under the center line of the arm, and the distance
to the resistance axis of the calculator, corresponding to zero reactance, would
then be the distance to the nearest point of minimum or maximum impedance. Here
the impedance is a pure resistance, and the distance to this point is read on the
distance scale in wavelengths. The value of this minimum or maximum resistance is
determined by moving the arm to this position and reading the value under the line
on the slide. The impedance at any other point, which might be the input end of
the line, is found by moving the arm the required distance and reading under the
intersection of slide and arm. If the distance were more than a half wavelength,
one or more half wavelengths would, before moving the arm of the calculator, have
to be subtracted from the total distance so that the remaining equivalent distance
will be less than a half wavelength.
At the load end of the line the impedance is Z_{r}, and this same value
of impedance will be found every half wavelength down the line as indicated in Fig.
4, where the successive points where Z = Z_{r} are marked. Between any two
such points there will be a point of minimum impedance, and one of maximum impedance.
At these points the impedance is a pure resistance. These points, which are represented
on the calculator by the transverse axis of zero reactance, are always a quarter
wavelength apart. The distance between them and any other point may be read from
the distance scale.
For most radio frequency transmission lines where the insulating medium between
conductors is chiefly air and the attenuation is negligible, the velocity of transmission
is practically the same as that of light. The distance scale around the edge of
the calculator is laid out in terms of wavelengths as measured on the transmission
line. For transmission lines having considerable attenuation and especially those
having an appreciable amount of high dielectric insulating material interposed between
conductors, there may be a marked reduction in the velocity of transmission, and
a correction factor must be applied to the distance scale. This correction is proportional
to the reduction in the velocity of transmission which can be measured or computed
from line constants.
To extend the utility of the calculator to audiofrequency telephone lines or
to other transmission lines, which may have an overall attenuation up to 15 decibels,
an attenuation scale expressed in 1 decibel intervals is provided along the adjustable
arm. The zero point along this scale will be variable and consequently, the scale
intervals are not numbered. For any particular problem, however, the zero point
will be found where the sliding crosshair on the arm intersects the scale when
the slider is first adjusted to any known transmission line impedance. If the problem
subsequently requires a movement of the rotatable arm along the distance scale "toward
generator," the attenuation scale will extend from this zero point in the direction
of the center of the calculator, or vice versa, as is indicated on the scale. It
is necessary only to count off the required number of decibels along the scale.
Moreover, if the input impedance at any two points along a transmission line is
known, the attenuation of the line can be obtained directly from this scale. This
will be indicated by the number of decibel intervals along this scale which the
slider moves across when going from one impedance setting to the other. It should
be noted, however, that for lines with reduced velocity and appreciable attenuation,
the characteristic impedance Z_{0} may be complex, and the complex value
should then be used to obtain the impedance quotients expressed as coordinates on
the calculator.
Posted June 21, 2023
