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.

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 impedance-matching
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.

An Impedance-Matching Transformer
Tutorial

A Simple Method for Matching the Antenna to the Transmission Line

By T.A. Gadwa, SC.D., W2KHM

Fig. 1 - Parallel-resonant circuit with equivalent
series and parallel resistances.

Fig. 2 - An impedance-matching circuit using series or tapped
inductances.

Fig. 3 - The pi-section filter, another type of impedance-matching
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 open-air two-wire
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 parallel-resonant 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 pi-section 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 two-wire line are shown in Figs.
5-C, 5-D, 6-C and 6-D. In Figs. 5-C and 6-C, one-half the total
inductance is put in each leg when the coils are not inductively
coupled. In Figs. 5-D and 6-D, 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 impedance-matching 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 2-element close-spaced 1/2-wavelength antenna at 14.2 Mc. to an
open-air parallel 2-wire line of No. 14 wire, with 6-inch 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 two-section stator with sections
in series is desirable because this construction eliminates losses
in rotor connections. For 300 watts through a 625-ohm 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 weather-tight box made of quarter-inch tempered Masonite, with
feed-through 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, quarter-wave
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 low-Q parallel circuit containing
a series resistance is broad enough to be used to advantage with
close-spaced antenna elements having a sharp frequency-response
characteristic. Its application is essentially to one-band 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 impedance-matching 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 parallel-resonant
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 parallel-resonant 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 antenna-to-line 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 half-wave 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 half-wave antenna at 114 Mc. to
an open-air 2-wire 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.

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