September 1948 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.

How RF circuits work have long been referred to as 'black magic,' even sometimes by people who fully understand the theory behind the craft. To me the ways in which a transmission line  be it coaxial cable, microstrip, or waveguide  can be manipulated and controlled with various combinations of lengths and terminations is what most qualifies as 'magic.' Sure, I know the equations and understand
(mostly) what's happening with incident and reflected waves, etc., and how the impedance and admittance circles of a Smith chart graphicaly trace out what's happening, but you have to admit there's something mystical about it all.
Antenna Matching with Line Segments
Design Formulas for WideRange Matching
By John G. Marshall, W0ARL
Although design charts for determining the length and position of a matching stub have been available for some time,
their use is restricted to the special case where the line and stub have the same characteristic impedance. This article
treats the linear matching transformer from a more general standpoint, giving considerably more latitude in the choice of
matching arrangements.
Many methods of matching the antenna to the transmission line have been described, but with the exception of the Qsection
transformer, very little design information has been published on those that employ a section of line as a transformer.
Practically nothing has been published on the actual design of the seriesbalanced network. The same holds true for the
shuntbalanced network, except for what has been written about the simplest form of the matchingstub system.
This article was prepared for the purpose of making available simple formulas for designing all types of networks that
employ a section of line as a transformer, whether series or shuntbalanced, including those in which the transformer section
and/or the stub, if used, have values of characteristic impedance different from that of the transmission line.
Early design of the matchingstub network consisted of connecting a λ/4 section of line to the antenna and attaching
the transmission line at the point that minimized the standing waves. In many cases, depending upon the ratio of antenna
drivingpoint impedance to transmissionline characteristic impedance, this procedure did not sufficiently reduce the standingwave
ratio. More recently, graphical solutions, which require the transmission line, transformer section and the stub itself
to have the same value of characteristic impedance, have appeared.^{1}
As will be seen, the formulas included here are not restricted in the above manner; and, if desired, each element of
the network may have a different value of characteristic impedance.
Since these networks employ the transformer action of a segment of line terminated by an impedance not equal to the characteristic
impedance, a brief review of line segments having lengths up to λ/4 is in order.1 Line segments possess many interesting
and valuable properties and those important to these networks are to follow.
When a section of line like that in Fig. 1 is terminated by a purely resistive impedance, Z_{L}, not equal to
the characteristic impedance, Z_{T}, the sendingend impedance, Z_{S}, contains a reactance component, X_{S},
as well as a resistance component, R_{S}, at all lengths, θ, except exact multiples of λ/4. Z_{S}
is actually the effective value of Z_{L} as seen through the section of line.
Input Reactance of Segment
Except when Z_{L} = Z_{T}, gradually increasing θ from zero causes the reactance, X_{S},
that appears at the sending end to rise gradually from zero to a maximum and then fall back to zero as θ reaches λ/4.
X_{S} is zero when θ is zero or λ/4, and is maximum when θ is one certain intermediate value.
This maximum becomes smaller as Z_{L} and Z_{T} approach equality, to the point where X_{S} is zero
at any value of θ when Z_{L} becomes equal to Z_{T}. The actual value of this maximum, or the value
of θ that causes it, is unimportant here. At lengths less than λ/4, X_{S} is inductive when Z_{L} <
Z_{T}; and when Z_{L} > Z_{T}, X_{S} is capacitive.
Input Resistance of Segment
Fig. 1  Simple transmission line segment.
When Z_{L} < Z_{T}, gradually increasing θ from zero causes the resistance, R_{S}, appearing
at the sending end to rise gradually from a minimum to a maximum, as θ reaches λ/4. The minimum value, which
is equal to Z_{L}, occurs when θ is zero, while the maximum value, which is equal to Z_{T}^{2}/Z_{L},
occurs when θ is &lambda/4. The greater the ratio Z_{T}/Z_{L} and the nearer θ is to λ/4,
the greater is the stepup transformer ratio.
When Z_{L}> Z_{T}, gradually increasing θ from zero causes R_{S} to drop gradually from
a maximum to a minimum, as θ reaches λ/4. This maximum, which is equal to Z_{L}, occurs when θ
is zero, while the minimum, which is equal to Z_{T}^{2}/Z_{L}, occurs when θ is λ/4.
The greater the ratio Z_{L}/Z_{T} and the nearer θ is to λ/4, the greater is the stepdown
transformer ratio.
A graphical representation of these effects is given in Fig. 2. Fig. 2A shows how the resistance and reactance vary
along a piece of 600ohm line terminated in 300 ohms, and Fig. 2B shows the variation along a 600ohm line with a 1200ohm
termination. The shapes of the curves would be the same for any similar ratios of Z_{L} and Z_{T}  only
the"Ohms" scale would change.
Irrespective of whether the transformer ratio is stepup or stepdown, as Z_{L} and Z_{T} approach equality
the smaller this ratio becomes. This may be carried to the point where Z_{L}, Z_{T}, maximum R_{S}
and minimum R_{S} all are equal. When this happens there are no standing waves, no X_{S}, and consequently,
a transformer ratio of 1 to 1 at any value of θ.
Fig. 2  Values of input resistance, R_{S}, and input reactance, X_{S}, at various segment
lengths. At (A) is shown a typical case where Z_{L} < Z_{T} (Z_{L} = 300, Z_{T} = 600),
and in (B) Z_{L} > Z_{T} (Z_{L} = 1200, Z_{T} = 600).
Fig. 3  Qsection transmission line transformer.
From the above, it is seen that a variety of transformer ratios is available by selecting various combinations of θ
and Z_{T}.
These curves are obtained from the relations
Since R_{S}, not X_{S}, handles the power, the transformer ratio between Z_{L} and R_{S}
is the heart of all antennamatching systems that employ the transformer action of a section of line. But in order to use
this transformer action θ must be fixed at some odd multiple of λ/4, unless some other means is provided to
balance out X_{S}.
Three general methods of treating the above reactive condition are illustrated in Figs. 3, 4 and 5.
QSection Transformer
Fig. 3 shows the popular Qmatch, which is covered in all the handbooks. It is briefly described here merely to show
its behavior and relationship to the other networks employing the linear transformer.
In this system, a λ/4 segment is selected having a value of Z_{T} that produces a Z_{S} containing
an R_{S} equal to the characteristic impedance, Z_{0}, of the transmission line. Since θ is an exact
multiple of λ/4, Z_{S} is purely resistive and there is no X_{S} to balance out.
With given values of Z_{L} and Z_{0},
Since Z_{T} is the only variable and there are limits to the useful range of characteristic impedances, the Qmatch
can be used to accommodate only part of the many combinations of Z_{L} and Z_{0} encountered.^{2}
As will be seen, the other networks employing the linear transformer are not limited in this respect.
SeriesBalanced Network
In the seriesbalanced network of Fig. 4, a segment is selected which has a convenient value of Z_{T} (usually
equal to Z_{0}) and of such length, θ, that Z_{S} contains an R_{S} equal to Z_{0}
In other words, the segment becomes a transformer having the proper ratio to make Z_{L} appear equal to Z_{0}
This condition is fully accomplished by balancing out the reactance component, X_{S}. by a series reactance, X_{BS},
of equal ohmic value but of opposite sign. Then the line looks into an impedance equal to its own Z_{0}.
With a suitable type of line selected
for the transformer section.^{2} the correct length, θ, and the total X_{BS} necessary to bring about
the above conditions, may be found from^{3}
and
When the same material is selected for the transformer section as for the transmission line  which is most common and
usually permissible  Z_{T} will equal Z_{0}, and simpler formulas may be used.^{2} In these cases,
formulas (1) and (2) reduce considerably, and values of θ and total X_{BS} may be found from
(3) and
X_{BS} = tan θ (Z_{L}  Z_{0}) ohms.
(4)
Fig. 4  Seriesbalanced network.
In the seriesbalanced network, the total X_{BS} should be equally divided between the two legs of the circuit.
It is important to note that when a capacitive balancing reactance is used each individual reactor must contain twice the
total capacity in order to contain half the total reactance.
The unmodulated peak voltage across each individual balancing reactor is
ShuntBalanced Network
In the shuntbalanced network of Fig. 5, a segment is selected which has values of Z_{T} and θ that render
a Z_{S} whose equivalent parallel impedance, Z_{P}, contains a parallel resistance component, R_{P},
equal to Z_{0}. The parallel reactance component, X_{P}, is balanced out by a parallel reactance, X_{BP},
of equal ohmic value but of opposite sign. Then the line looks into a pure resistance equal to its own Z_{0}.
With a suitable type of line selected for the transformer section,^{2,3} the correct values of segment length, θ,
and parallel balancing reactance X_{BP}, necessary to bring about the above conditions may be found from
and
As in the seriesbalanced network, if the same material is selected for the transformer section as for the transmission
line, Z_{T} will equal Z_{0} and simpler formulas may be used.^{2} In these cases, formulas (5)
and (6) reduce considerably, and values of θ and X_{BP} my be found from
The unmodulated peak voltage across X_{BP} is
Table I  Proper Formulas for Finding Length of Transformer Section and Value of Balancing Reactance
* When a stub is desired at X_{BP}, β is found from (9) or (10).
Linear Shunt Reactors
The shuntbalanced network is especially suited to the use of a linear balancing reactor, such as that made of a segment
of open or closed line. It is quite convenient that any practical value of characteristic impedance, Z_{C}, may
be selected for the linear reactor or stub.
After selecting a value for Z_{C}, the necessary length, β, to give the required value of X_{BP},
may be found from
(9)
and
(10)
for the open and closed stub, respectively.
The Handbook^{1} shows that when β is less than λ/4, an open stub is a capacitive reactance while
a closed stub is an inductive reactance. Formulas (9) and (10) bear this out.
NOMENCLATURE
Z_{0}  Characteristic impedance of transmission line
Z_{T}  Characteristic impedance of transformer section
θ  Length of transformer section
Z_{C}  Characteristic impedance of stub
β  Length of stub
Z_{L}  Impedance of antenna driving point (must be nonreactive)
Z_{S}  Sendingend impedance of transformer section
R_{S}  Resistance component of Z_{S}
X_{S}  Reactance component of Z_{S}
X_{BS}  Series balancing reactance
Z_{P}  Parallel equivalent of Z_{S}
R_{P}  Resistance component of Z_{P} X_{P}  Reactance component of Z_{P} X_{BP}
 Parallel balancing reactance
E_{S}  Voltage across series balancing reactor
E_{P}  Voltage across parallel balancing reactor
W_{O}  Power output of transmitter
V  Velocity factor
MatchingStub Network
In the matchingstub network, which is a special form of the shuntbalanced network, it is convenient and most common
practice (although not essential) to construct the transformer section, transmission line and balancing reactor from the
same material.^{2} When this is done Z_{T}, Z_{0} and Z_{C} are equal, and design formulas
become quite simple. Once a line of known Z_{0} has been selected, it is necessary to find only θ and β.
Since this system is of the shuntbalanced type and Z_{T} = Z_{0}, tan θ is found from formula
(7).
When Z_{L} < Z_{0}, an open stub is used and formulas (8) and (9) combine into one operation. Then β
may be found from
When Z_{L} > Z_{0}, a closed stub is used and formulas (8) and (10) combine into one operation. Then
β may be found from
Examples
Table. I will aid in selecting the proper formulas to use in working any example using any of these networks.
In working an example, it is necessary to convert degrees to feet. A useful formula, requiring a minimum of effort, is
where V is the velocity factor of the line.
Fig. 5  Shuntbalanced transmission line network.
Example 1 
Given: A matchingstub network with Z_{L} = 70 ohms, Z_{0} = 600 ohms, and V = 0.975, operating on 7
Mc.
Solution: Needed are θ and β. According to Table I, an open stub with formulas (7) and (11) is used. Then,
and
From trig tables, θ = 18.9° and β = 68.9°. Converting to feet via formula (13), θ = 7.19 feet
and β = 26.2 feet.
Example 2 
Given: A shuntbalanced network with Z_{L} = 8 ohms, Z_{0} = 75 ohms (TwinLead), V = 0.71, and W_{O}
= 1 kw., operating on 14.1 Mc.
Solution: Needed are θ and X_{BP}. With due consideration for Footnote 2, it is decided not to use the
75ohm TwinLead in the transformer section, since the power is high and Z_{L} and Z_{0} are quite different.
To assure a minimum of losses, 1inch tubing spaced 1 1/2 inches is tried.^{3} This has a Z_{T} of 150 ohms
and an estimated V of 0.95. According to Table I, formulas (5) and (6) are used. Then
and θ = 9.0° which, when converted to feet, eauals 1.66 feet.
and when converted to capacitance equals 431 μμfd. Note that in these networks the value of Z_{T} does
not have to be between the values of Z_{L} and Z_{0} Z_{T} may be of any value that complies with
the requirements of Footnote 3.
Summary
Engineering handbooks give formulas for finding the sendingend impedance of a segment of line having any value of terminating
impedance. Typical of these is
From this basic equation, the network formulas in this paper were derived.
From the standpoint of efficiency, there is little choice between the three general systems treated here. Because of
its simplicity, the Qsection is the logical choice when the necessary value of Z_{T} is within the practical range
of characteristic impedances mentioned earlier."
The importance of having a purelyresistive driving point in the antenna is stressed. As in other types of networks,
any appreciable amount of reactance (as compared with the resistance of Z_{L}) will cause standing waves to appear
on the transmission line. The driven element should be selfresonated before attaching the network.^{2}
With the aid of the formulas included here, a network having a minimum of losses can be designed to accommodate about
any conceivable combination of antenna and transmissionline impedances. It is hoped they will be helpful.
1. Radio Amateur's Handbook, antenna chapter.
2. There is another consideration important to the Q as well as to all other networks employing the transformer action
of a segment of line. When Z_{L} and Z_{0} differ greatly. the standingwave ratio is high and the use of
soliddielectric cable in this section may result in considerable power loss or possibly breakdown. Cables are rated under
flatline conditions and the maximum rated r.m.s, voltage is where W is the rated power and Z is the characteristic impedance.
The voltage at the antenna end of the transformer section in any of these networks is The voltage at the sending end of the Q section and the shuntbalanced
network is In the seriesbalanced network it is where I_{L} is the current at the antenna and equal
to When Z_{L} < Z_{0} maximum voltage is at the
sending end in any of these networks, while when Z_{L} > Z_{0} maximum voltage is found at the antenna
end.
3. A negative quantity appearing under the radical in formulas (1) and (5) indicates that the value of Z_{T}
selected does not permit sufficient transformer ratio, even if θ is made the full λ/4, so another selection
must be made. To be workable, Z_{T} must be greater than
EQUATION HERE
when Z_{L} < Z_{0}, and Z_{T} must be less than
EQUATION HERE
when Z_{L} > Z_{0}
4 For methods of resonating the driven element, see Potter, "Establishing Antenna Resonance," QST, May, 1948, and Smith,
"Adjusting the Matching Stub," QST, March, 1948.  Editor
Posted April 14, 2016
