
An ImpedanceMatching Transformer Tutorial
February 1943 QST Article


February 1943 QST
[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. 
The
word 'transformer' in the title for this article does not refer to a
mutual inductance transformer, but an impedance transformer for matching
transmission lines to antennas (or anything else for that matter). Author
T.A. Gadwa gives examples of impedancematching circuits both for
when the antenna impedance is lower than the characteristic impedance
of the transmission line and when the antenna impedance is higher than
that of the feed line. "L," "pi," and a couple other circuit configurations
are covered.
See all available
vintage QST articles.
An ImpedanceMatching Transformer Tutorial
A Simple Method for Matching the Antenna to the Transmission Line
By T.A. Gadwa, SC.D., W2KHM
Fig. 1  Parallelresonant circuit with equivalent
series and parallel resistances.
Fig. 2  An impedancematching circuit using series or tapped
inductances.
Fig. 3  The pisection filter, another type of impedancematching
circuit.
Fig. 4  These circuits resemble those of Fig. 2 with L and
C interchanged. 
While those of us at home don't have many opportunities these days to
try tuning up antenna systems, the method described in this article
will some day be useful to us. At present, it can be applied to WERS
communication, design data for a suitable coupler being included in
the article.
Any simple and inexpensive method of coupling an
antenna to a transmission line always is attractive to amateurs. Numerous
articles on untuned feeders have outlined their advantages  lower losses,
reduced feeder radiation and operation independent of line length. An
antenna placed in a favorable location and supplied power by untuned
feeders or transmission lines is frequently desirable, but coupling
one end of the transmission line to the plate circuit and the other
to the antenna does not solve the problem satisfactorily. To transfer
power most efficiently on such a transmission line, the load resistance
must equal the generator resistance. This means that power is absorbed
by the load and none is reflected back to the sending end to produce
standing waves. If the termination differs from this load resistance,
standing waves appear on the line, representing wasted power that never
reaches the antenna. The character of standing waves for various types
of loads has been described previously^{1} and may be reviewed
for reference purposes.
A transmission line of two parallel
conductors has a characteristic impedance which is determined by the
physical dimensions of the system: diameter of the conductors, their
spacing and the insulation or dielectric. The equation for calculating
the impedance of an openair twowire parallel line is:
R_{o}
= 276 log 2S/D (1)
where R_{o}
= characteristic impedance of the line in ohms S = spacing between
conductor centers in any units D = diameter of conductor in same
units
Impedance Transformation
In come cases, the impedances of an antenna and transmission line
are not equal and some sort of transformation must occur before the
load can be matched to the line. It is possible to convert an impedance
to a higher or lower value by utilizing a circuit known as a filter,
network or impedance transformer, composed only of inductances and capacitances.
When a filter of suitable design is inserted between the antenna and
transmission line, the load presented to the line will be equal to the
line impedance, and an impedance match for a flat line is possible.
A parallelresonant circuit of inductance, capacitance and resistance,
such as is shown in Fig. 1, has different impedances between various
points of the circuit. The impedance between any two points can be found
by combining the series and parallel elements in the usual manner. A
pisection filter will accomplish the same transformation, which is
equivalent to tapping the antenna across a portion of the inductance
or capacitance. These arrangements, shown in Figs. 2, 3 and 4, are not
recommended since they require one more element than the circuit of
Fig. 1; also, it is impossible to obtain a correct impedance transformation
for certain combinations of inductance and capacitance because of insufficient
coupling. The impedance transformer should exhibit pure resistance at
its terminals, and Everitt^{2} has shown what the values of
the inductive and capacitive reactances should be to satisfy this condition.
Equations which have been used in previous QST articles,^{3, 4, 5}
are:
where
X_{L} = inductive reactance in ohms X_{C}
= capacitive reactance in ohms R_{1}
= input or output resistance R_{2}
= output or input resistance L = X_{L}/2πƒ (4)
C = 1/2πƒX_{C} (5) ƒ = frequency in cycles
per second L = inductance in henrys C = capacitance in farads
A resonant antenna can be connected to one pair of
terminals and its effective impedance at the second pair of terminals
changed to equal that of the line. The antenna behaves like a series
resonant circuit and is a pure resistance at resonance. It is reactive
off resonance  capacitive at frequencies below resonance and inductive
at frequencies above resonance. For the case where the resistance of
the antenna is lower than that of the transmission line, the circuits
in Fig. 5 can be employed. Circuits in Fig. 6 are used when the antenna
resistance is higher than the line impedance. Symmetrical arrangements
of the circuits for connection to a twowire line are shown in Figs.
5C, 5D, 6C and 6D. In Figs. 5C and 6C, onehalf the total inductance
is put in each leg when the coils are not inductively coupled. In Figs.
5D and 6D, one fourth the total inductance (half the total number
of turns) is put in each leg when the coils are inductive coupled.
To aid in the solution of equations (2) and (3) curves are presented
in Figs. 7 and 8. From the inductive and capacitive reactances, the
inductance and capacitance can be determined from equations (4) and
(5). From the inductance, the coil diameter, length of winding and number
of turns may be found by the usual formulas or from a Lightning Calculator.
Fig. 5  These impedancematching circuits are
used when the antenna resistance is lower than the characteristic impedance
of the transmission line.
Fig. 6  Circuits for use when the antenna resistance
is higher than the line impedance.
A Practical
Example
To illustrate the various steps in the calculation,
a typical case is solved. It is desired to match the resistance at the
center of one element of a 2element closespaced 1/2wavelength antenna
at 14.2 Mc. to an openair parallel 2wire line of No. 14 wire, with
6inch spacing between wires. The characteristic impedance of the line
is obtained from equation (1).
R_{o} = 276 log (2 X
6/0.064) = 276 log (188) = 276 X 2.275 = 625 ohms
The antenna
resistance may be assumed to be equal to 13 ohms. Since the line impedance
is higher than the antenna resistance, a transformer of type shown in
Fig. 5 must be employed. The inductive reactance from equation (2a)
is X_{L} = 13√(625/13 1) = 13 x 6.86 =
89.2 ohms
The required inductance, from equation (4) is
or 1.00 microhenry. Using the Handbook formula
where N = number of turns
A = diameter of coil in inches
(let A = 1. 5 inches) B = length of coil in inches (let B = 1.5
inches) L = inductance in microhenrys
Within small limits, the inductance can be increased by spacing
the turns closer together and decreased by spacing them farther apart.
Antenna material is satisfactory for the coil, although heavier wire
or copper tubing will keep the losses to a minimum.
The capacitive
reactance, from equation (3a) is X_{C} = 625 / √(625/13
1) = 625 / 6.86 = 91.1 ohms
The required capacitance, from
equation (5), is
or 123 micromicrofarads. The voltage across the condenser is relatively
low because of the low impedance involved. Receiving type condensers
are satisfactory, since the plate spacing need not be large for most
amateur powers. A twosection stator with sections in series is desirable
because this construction eliminates losses in rotor connections. For
300 watts through a 625ohm line, the voltage is
E = √(PR)
= √(300 x 625) = 433 volts r.m.s.
Fig. 7  Parallel resistance vs. inductive reactance for various
values of series resistance.
Fig. 8  Parallel resistance vs. capacitive reactance for various
values of series resistance.

The peak is 433 X 1.414 = 610 volts and on 100 percent modulation the
peak is 610 X 2 = 1220 volts.
The tuning unit must be protected
from the weather. One version of such an impedance transformer is illustrated
in the photograph. The coil and condenser are mounted in a weathertight
box made of quarterinch tempered Masonite, with feedthrough terminals
brought out through the sides for the line and similar terminals at
one end for the antenna.
Interference with the antenna radiation
field by matching stubs, quarterwave sections and delta matching sections
are avoided when the transformer is used, since the transformer is concentrated
in a much smaller space. The frequency response of such a lowQ parallel
circuit containing a series resistance is broad enough to be used to
advantage with closespaced antenna elements having a sharp frequencyresponse
characteristic. Its application is essentially to oneband antennas
since impedance transformation is dependent upon the frequency of operation.
It must be emphasized that one and only one combination of inductance
L and capacitance C will match a given antenna resistance to a given
line. As the ratio R_{2}/R1 approaches unity, X_{L}
approaches zero and X_{C} approaches infinity; that is, the
inductance and capacitance both become smaller. The resonant frequency
of L and C without R_{1} may be considerably higher than with
R_{1} in the circuit.
Adjustment
It is highly desirable to be able to tune the unit when it is
in its operating position at the antenna. This may be done by varying
the capacity until maximum antenna current is shown by an r.f. ammeter
or lamp bulb connected in the antenna at the junction to the transformer.
Alternatively, one may adjust for minimum line current at the line junction
to the impedance transformer. Where this is impossible or inconvenient,
it is permissible to tune the coil and condenser to resonance before
connecting the antenna and transmission line. Since the resonant frequency
of the coil and condenser alone always is higher than with the antenna
in the circuit, the capacity is then reduced sufficiently to compensate
for the insertion of the antenna when the unit is in operating position.
If the antenna is resonant and the correct values of inductance and
capacitance are employed, the line will be correctly terminated. A constant
current at all points along the line, or a slight increase of current
toward the transmitter or sending end, is the final test of a perfect
impedance match.
A thermomilliammeter connected across a portion
of one feeder line at various positions is a good indicator of standing
waves. A flashlight bulb connected across a short length of one feeder
is also a good current indicator and is inexpensive. The bulb should
be shielded to direct the light to the observer so that the neighbors'
curiosity will not be aroused by night operation. If bulbs are permanently
located at intervals of 1/16 wavelength along the line, starting from
the antenna, the brilliancy vs. position shows the location of maximum
and minimum line currents or standing waves.
Suggested construction of an impedancematching transformer
for suspension from an antenna. The condenser is controlled
by the arm projecting toward the upper left. A pulley could
he used for adjustment from the ground. 
If the antenna is nonresonant, its length must be adjusted or tuned
to resonance. Excite the antenna parasitically and obtain maximum antenna
current by tuning. Noting the position of the standing waves on the
transmission line, as outlined in the article on standing waves,^{1}
also is recommended. One exception must be observed because the resistance
across the terminals of a parallelresonant circuit increases when the
series resistance decreases. In other words, the load resistance presented
to the line is increased for a decrease in antenna resistance and, conversely,
the load resistance presented to the line is decreased for an increase
in antenna resistance. This may be understood by analyzing the approximate
relationship that holds for a parallelresonant circuit of low series
resistance or high Q.
R_{2} = L / (C x R_{1})
This is true when R_{1} is relatively small and is approximately
so for higher values of R_{1}. It means that the parallel impedance
is increased by using larger inductance L and a smaller capacitance
e (increasing L/C ratio), and by reducing the series resistance R_{1}.
Conversely, the parallel impedance is decreased by using a smaller inductance
L and a larger capacitance e (decreasing L/C ratio) and by increasing
the series resistance R_{1}. The parallel resistance always
is greater than the series resistance.
If the antenna is resonant
but incorrect values of inductance and capacitance are used in the impedance
transformer, a current loop or node will appear near the 1/4 wavelength
point measured along the line from the transformer. If a current loop
or maximum occurs at this position the terminating resistance is too
high, and a smaller inductance L and a larger capacitance C are required.
If a current node or minimum occurs near the 1/4 wavelength position,
the terminating resistance is too low and a larger inductance L and
lower capacitance C are required.
If the antenna and line resistances
are known, the ratio of the line and antenna currents for an impedance
match can be calculated from the square root of the antennatoline
resistance ratio. This is based upon the assumption that the power input
to the transformer equals the power output; i.e., that the losses in
the transformer are negligible.
P = I^{2}R = I_{1}^{2}R_{1}
= I_{2}^{2}R_{2} or
I_{1} /
I_{2} = √(R_{2} / R_{1})
If an
r.f. ammeter is available, measurement of the antenna and line currents
will reveal the correct impedance match from their ratio.
Fig. 9  This circuit is used for matching a halfwave antenna
to a line having an impedance of the order of 500 ohms. Constants
for 114 Mc. operation are given in the text. 
With 2 1/2 meters active for civilian defense, transmitting antennas
and associated problems are under consideration once again. A design
is given in Fig. 9 for matching a halfwave antenna at 114 Mc. to an
openair 2wire line of No. 14 wire spaced 2 inches:
S = 2 inches
spacing D = 0.064 wire diameter, inches R_{2} = 495
ohms, line impedance R_{1} = 73 ohms, antenna resistance
ƒ = 114 X 10^{6} cycles per second X_{L}
= 175.8 ohms L = 0.245µh. A = 1 inch (coil diameter)
B = 1 inch (coil length) N = 3.8 turns N/2 = 1.9 turns
X_{C} = 206 ohms C = 6.8 µµfd.
It
is hoped that this method will not be overlooked when considering the
problem of matching the antenna to the transmission line. Because of
its simplicity, it might well be adopted by the amateur radio fraternity.
^{1}
Gadwa, "Standing Waves on Transmission
Lines," December, 1942, p. 17. ^{
2} Everitt,
Communication Engineering, p. 75. ^{3} Andrews, QST, October,
1939, p. 39. ^{4} Plotts, QST,
November, 1941, p. 15. ^{5} Roberts, QST, January, 1928,
p. 43.
Posted 6/13/2013



