[Table
of Contents] These articles are scanned and OCRed from old editions of the
ARRL's QST magazine. Here is a list of the
QST articles I have already posted. All copyrights (if any) are 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 well-established
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 cross-platform Java software called
SimSmith. If you
want to do a little complex number math, try the 1966 approach.
See all available
vintage QST articles.
L Networks for Reactive Loads
By Robert E. Gordon, W0KFI/ex-W1KUL
Calculation for Matching
Antenna System to Transmitter
Two L networks designed by the author. To the right, the 3950-kc.
network sketched in Fig. 4 is shown enclosed in a coffee can.
The 14-Mc. network to the left was photographed before mounting
in a similar shielding enclosure.
Fig. 1 - L-network 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 L-network formulas for transforming an antenna-system 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 50-ohm resistive load my transmitter prefers.
The Handbook formulas for the design of L networks are limited to
cases of transforming pure resistances. Unfortunately, a feed-point
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 50-ohm 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 Step-Down 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 50-ohm 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 1000-and a 200-pf. mica capacitor, and a piece
of 5/8-inch 16-pitch 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 previously-reluctant 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 Step-Up
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 impedance-matching 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 coffee-can job described earlier, together with a prototype
20-meter 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 NCX-5 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