January 1954 RadioElectronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from RadioElectronics,
published 19301988. All copyrights hereby acknowledged.

I didn't know that
another name for a nomogram (or nomograph) is an "abac," but thanks to Mr. Crowhust
in this 1954 RadioElectronics magazine article I now do. At first I
thought maybe it was a shortened version of abacus, but I couldn't find anything
suggesting that. Anyway, this article presents a couple abacs you do not often
(if ever) see  the attenuation and reflection levels for resistive attenuators
when the impedance(s) are incorrect for the designed values. Keep in mind,
particularly when using offtheshelf components like attenuators, filters,
power couplers and splitters, PIN switches, etc., that if your source and load impedances are not
exactly the same as the component input and output impedance (usually 50 Ω or
75 Ω), the response will not be as advertised. Achieving the proper
performance will require inserting an impedance matching section between the
circuit and the component.
Maintaining proper attenuation and impedance reflection when connecting
two different impedances.
Chart I  Reference line in the middle gives the ratio between
the impedances and attenuation resulting from mismatch.
By N. H. Crowhurst
In the preceding article (page 36, December) on the design of constant impedance
attenuators, the author stated that attenuators perform correctly only when properly
terminated. If terminated with impedances other than those on which the design is
based, both attenuation and impedance reflection through the attenuator become incorrect.
Two of the three charts here deal with the result of using an attenuator of the
constantimpedance type with incorrect impedances, while the third shows how to
design matching pads for connecting together two different impedances so that both
terminating impedances are matched.
Chart I is a simple alignment abac (alignment chart or monogram) with two impedance
scales on the outside and a reference line in the middle, giving on the left, the
ratio between the impedances on the outside scales, and on the right, the mismatch
loss resulting from connecting these impedances together.
If a constantimpedance attenuator is connected to one of the working impedances,
and is correctly designed for that impedance, the presence of the attenuator will
not affect the mismatch loss, because the impedance reflected to the opposite terminals
will be the same as the correct impedance terminating it. This applies to only those
attenuators, either where the constant impedance characteristic applies for both
directions, or, in the case of L pads, where the attenuator is correctly terminated
in the direction for which it is designed to have constant impedance. It would be
pointless to extend the chart for other cases using L pads, because there seems
to be no reason why the correct matching type L pad could not be used.
Chart II  For determining mismatch reflection through the attenuator
when attenuator is improperly terminated.
Chart III  To determine minimum attenuation necessary for correct
matching, and resistor values required for Lpads.
The mismatch loss, over and above the attenuation of the attenuator, will be
quite independent of either the type of attenuator or the amount of attenuation
for which it is designed.
Chart II deals with the mismatch reflection through the attenuator. The diagram
on page 119 shows that the attenuator is designed for a working impedance Z_{0}
and a specific attenuation in db. It is, however, improperly terminated by the impedance
Z_{2}. As a result, the chart shows what impedance will appear at the opposite
end of the attenuator, represented by the symbol Z_{2'}. The type of attenuator
is unimportant. What affects the mismatch reflection is the design figure of attenuation
for the particular unit.
It is also unimportant whether the designed attenuation is achieved in one stage
or in a number of separate stages. For example, the mismatch reflection would be
the same through 40 db of constantimpedance attenuation, whether this was made
up in a single pad, or of a number of separate pads of, say, 20 plus 10 plus 5 plus
2 plus 2 plus 1.
Example 1
A constantimpedance attenuator designed for 5,000 ohms is terminated by a resistance
load of 5 ohms resulting in a mismatch loss.
To find the mismatch loss: Chart I, using Z_{2} 5,000 ohms and Z_{1}
5 ohms, gives 24 db. Using Chart II to find the mismatch reflected impedance, Z_{2}/Z_{0},
as given from Chart I, is 0.001, which will be indistinguishable from zero at the
righthand side of Chart II. This means, that, to all intents and purposes, a 5ohm
resistance connected to a 5,000ohm attenuator is a short circuit. Suppose the attenuation
is designed for 20 db, then by using the straightedge, or ruler, as shown in the
key on Chart II, Z_{2'}/Z_{0} is given as 0.98, which means the
input impedance is 0.98 x 5,000 ohms, or 4,900 ohms.
If the attenuation was 10 db, Z_{2'}/Z_{0} becomes 0.82, making
the impedance at the input end of the attenuator 4,100 ohms.
Example 2
An attenuator is designed for 250 ohms and is actually terminated by 600 ohms.
Using Chart I, the ratio Z_{1}/Z_{2} is 2.4, and the mismatch loss
attenuation is slightly less than 1 db.
Turning to the mismatch impedance transfer, using 2.4, Z_{2}/Z_{0},
which is in the left hand part of the arc on Chart II, and supposing the attenuator
is designed for an attenuation of 6 db, alignment gives a reading of 1.24 for Z_{2'}/Z_{0}.
So, the reflected impedance is 1.24 x 250, or 310 ohms.
If the attenuation is 20 db, the mismatch drops to a ratio of 1.01, representing
252.5 ohms.
To avoid any trouble due to mismatch, which in r.f. can result in reflections
and their consequent troubles as well as just losses (as in television attenuators),
it is advisable to design a matching pad so that both impedances are correctly matched.
This can be done quite easily with Chart III.
The chart gives the minimum attenuation necessary to produce correct matching,
and the resistor values for an Lpad to produce such matching. This pad can of course
be combined with further attenuators of the constantresistance type, designed at
either of the working impedances, using one of the charts in the earlier article.
To use Chart III, the two impedances to be matched, denoted by Z_{1}
and Z_{2}, are aligned; and the db attenuation can be found on the lefthand
side of the center reference line. The figure from this scale is then transferred
to the scale on the righthand side of the same reference line, and aligned with
Z_{1} and Z_{2}. This locates the values of R_{1} and R_{2}
as shown in the key on the chart.
Example 3
Take the case of 5,000 ohms being matched down to 5 ohms. To do this correctly,
using Z_{1} 5,000 ohms and Z_{2} 5 ohms, the attenuation
is 36 db. On the righthand center scale, 36 db is so close to the infinity
point as to be indistinguishable from it. In aligning 5,000 ohms with this
point to find the value of R_{1} the straightedge falls beyond the bottom
of the righthand line; similarly for the value of R_{2}. This can be overcome
by dividing by 10 on the Z scales; then converting the R value back again by multiplying
by 10. Aligning Z_{1} at 500 with the infinity mark gives R_{1}
as 500, which means its actual value will be 5,000 ohms; similarly R_{2}
is 5 ohms.
Now consider the case where the impedances are closerthe second example given
above  working between 250 and 600 ohms. Using Z_{1} as 600 and Z_{2}
as 250, the attenuation is about 8.8 db. Transferring this to the righthand reference
scale on the center reference line, R_{1} is 440 ohms and R_{2}
is 325 ohms.
Values as calculated from the chart are the exact values required for perfect
matching. In practice, the closest preferred value resistor in the 5% or 10% range
will probably give a good enough match for all practical purposes. It will almost
certainly result in obtaining much closer matching than using an attenuator without
giving proper attention to matching. In most cases 10% resistors will do.
Posted January 28, 2022
