September 1966 QST
Table
of Contents
Wax nostalgic about and learn from the history of early electronics. See articles
from
QST, published December 1915  present (visit ARRL
for info). All copyrights hereby acknowledged.

"L" networks are probably the most
common types of impedance matching networks not just for antennas, but for an relatively
narrowband load. Determining the required values for the network is relatively simple
using wellestablished equations. Knowing how to use a Smith Chart makes the job
even easier. This article from a 1966 edition of QST magazine presents
the equation approach. If you have access to the May 2013 edition of
QST, there is a complimentary article on L networks that uses the free
Smith Chart crossplatform Java software called
SimSmith. If you want to do a little complex number math, try
the 1966 approach.
L Networks for Reactive Loads
Two L networks designed by the author. To the right, the 3950kc. network sketched
in Fig. 4 is shown enclosed in a coffee can. The 14Mc. network to the left
was photographed before mounting in a similar shielding enclosure.
Fig. 1  Lnetwork configurations, (A) for stepping up the input impedance,
(B) for stepping the input impedance down.
By Robert E. Gordon, W0KFI/exW1KULCalculation for Matching Antenna System
to Transmitter
The usual Lnetwork formulas for transforming an antennasystem impedance to
a value appropriate for the transmitter assume that the antenna impedance is a pure
resistance. The author observes that this condition seldom occurs in practice, and
proceeds to discuss the more prevalent case of a complex antenna load.
An article by W8CGD in an earlier issue of QST^{1} describes an inexpensive
device for measuring antenna or other complex impedances, with ample accuracy for
most purposes. I have made use of it in designing L networks to transform odd antenna
impedances to the 50ohm resistive load my transmitter prefers.
The Handbook formulas for the design of L networks are limited to cases of transforming
pure resistances. Unfortunately, a feedpoint impedance which contains no reactive
component is about as rare as a dodo. Accordingly, I derived formulas for transforming
any load impedance to a pure resistance of any desired value.
The L network has two possible configurations. When the resistive component of
the load is greater than the desired generator resistance (R_{O}>R_{I}),
the parallel element will be on the load side, as shown in Fig. 1A. Conversely,
when the resistive component of the load is less than desired generator resistance
(R_{O}<R_{I}), the parallel element will be on the generator
side, as shown at B. In the case of a load resistance equal to the desired generator
resistance it is not necessary to use the formulas, since it is apparent that compensation
will be required for the reactive component only and this may be obtained by a single
series clement having the same numerical value of reactance as that contained in
the load, but of the opposite sign. For example, if we wish to present a 50ohm
resistive load to the transmitter, and the antenna impedance measures 50  j30 (capacitive),
we would place an inductor of reactance +j30 in series with the antenna. The transmitter
will now see 50  j30 + j30, or simply 50 ohms, resistive.
The formulas for the two networks of Fig. 1 are different, so we will look
at them one at a time, and work out an example for each. In all of these formulas,
the subscript I refers to the input resistance of the network, O to the output impedance
of the network, and S and P to the series and parallel network reactances, respectively.
Hence, if we are trying to match a transmitter to an antenna, R_{I} represents
the desired resistive load we wish to present to the transmitter, and R_{O}
+ jX_{O} represents the actual antenna impedance which we have measured.
Two L networks designed by the author. To the right, the 3950kc. network sketched
in Fig. 4 is shown enclosed in a coffee can. The 14Mc. network to the left
was photographed before mounting in a similar shielding enclosure.
Fig. 1  Lnetwork configurations, (A) for stepping up the input impedance,
(B) for stepping the input impedance down.
Fig. 2  Formulas and configuration for the case where the resistive component
of the load impedance is smaller than the desired load for the transmitter.
Fig. 3  Inductive and capacitive elements in an L network may be transposed
with suitable changes in values, as discussed in the text. Values shown here are
for the author's case of transforming a measured 17  j6.5 antenna load at 3950
kc. to 50 ohms resistive for the transmitter.
Fig. 4  Sketch showing the construction of the network of Fig.3A.
Fig. 5  Configuration and formulas for an L network for the case where
the resistive component of the load impedance is larger than the desired load for
the transmitter.
The StepDown L Network
We will start with the case where the resistive part of our load (R_{O})
is less than the desired input resistance (R_{I}). See Fig. 2 for
the network sketch and formulas. The factor A has been introduced to simplify the
arithmetic. My transmitter, which is designed to operate into a 50ohm resistive
load, would not tune up to the antenna on 3950 kc. Measurement on the antenna using
W8CGD's device showed the reason: a measured impedance of 17  j6.5.
Here is how we proceed to design the required L network:
R_{I} = 50 ohms
R_{O} = 17 ohms
jX_{O} = j6.5 ohms
1)
2) jX_{S} =  jX_{O} + jR_{O}A
= j6.5 + j(17)(1.393)
= j30.2
3) The plus sign tells us that the required reactance is inductive.
The inductance required to yield a reactance of 30.2 ohms at 3950 kc. is then:
4)
5) The minus sign tells us that this reactance is capacitive. The capacitance
required to provide a reactance of 35.9 ohms at 3950 kc. is:
The circuit is then as shown in Fig. 3A.
Now, on to the junk box. It produced a 1000and a 200pf. mica capacitor, and
a piece of 5/8inch 16pitch coil stock. The Handbook graph says 10 1/2 turns of
this will come pretty close to 1.23 µh., and the capacitance is pretty close
to what we need. Adding one coffee can, coax and connectors, and a couple of hours
in the cellar, produced the object shown in the sketch of Fig. 4 and the photo.
With this network patched into the antenna lead, the previouslyreluctant transmitter
now loads without difficulty from 3900 to 4000 kc,
Before leaving this topic, it should be mentioned that there is also another
pair of reactance values which would do the same job if the inductive and capacitive
elements are transposed. The values required may be computed in the same manner
as given in the example, but using these formulas:
jX_{S} =  jX_{O}  jR_{O}A
jX_{P} =  jR_{I}/A
where A has the same meaning indicated earlier.
Using the data of the foregoing example, these formulas yield results as follows:
jX_{S} =  j17.2 C_{S} = 2350 pf.
jX_{P} = j35.9 L_{P}
= 1.44 µh.
The circuit is as shown in Fig. 3B.
A network using these values would have performed equally well, but the required
components are larger, and the internal d.c. ground on the coax center conductor
found in most transmitters would be blocked from the antenna by the series capacitor.
It may be worthwhile to figure the values both ways, and choose the arrangement
you like best.
The StepUp L Network
The other network configuration (Fig. 1A) must be used when the resistive
part of our load (R_{O}) is greater than the desired input resistance (R_{I}).
See Fig. 5 for the network sketch and formulas. None of my antenna measurements
produced values of R_{O} greater than the desired R_{I}, so I have
invented some values, for the purpose of an example, as follows:
R_{I} = 50 ohms
R_{O} = 70 ohms
jX_{O} = +j20 ohms
ƒ = 14.1 megacycles
1) Z_{O}^{2} = R_{O}^{2} + X_{O}^{2}
= (70)^{2} + (20)^{2}
= 4900 + 400 = 5300
2)
3)
4)
(It will be noticed that j was shifted from the denominator to the numerator
with a change of sign. This is accomplished by multiplying both numerator and denominator
by  j.)
5)
As in our previous case, there is another pair of values which will also do the
same job, obtainable by the following formulas:
Again using our same data, these formulas yield results as follows:
jX_{S} =  j35.8
jX_{P} = j176
C_{S} = 317 pf.
L_{P} = 1.99 µh.
A network using these values would do the same impedancematching job as the
preceding one.
As a concluding comment applicable to both network configurations, I would point
out that in some cases both the series and parallel elements will be of the same
kind (L or C), so if you come out with this result it doesn't necessarily signal
an error in arithmetic.
The photograph shows the coffeecan job described earlier, together with a prototype
20meter job which, not being canned, is more photogenic. It has since been canned
to reduce undesired local radiation. This one, you will notice, required two inductors.
The companion set of formulas yielded an LC combination, but Miniductor is a lot
easier to trim to size than a molded mica brick.
These networks have been wholly successful in enabling me to feed my NCX5 transceiver
into a trap dipole, with plenty of room to spare on the transmitter adjustment,
where previously it had been impossible to achieve the manufacturer's recommended
conditions of loading.
I should like to acknowledge the many helpful suggestions of Doyle Strandlund,
W8CGD, during the preparation of this article.
Now with a few hours of effort, you can really transform the needles, noodles,
and wet string to 50 + j0. Who will be the first to build an d. noodle drier?
1 Strandlund, "Amateur Measurement, of R + jX," QST, June, 1965.
Posted February 25, 2022 (updated from original post on 6/3/2013)
