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L Networks for Reactive Loads
September 1966 QST Article 
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 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
By Robert E. Gordon, W0KFI/exW1KUL
Calculation for
Matching Antenna System to Transmitter
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 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.
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 6/3/2013

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