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September 1948 QST Article

## September 1948 QSTThese 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 are 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.

Design Formulas for Wide-Range 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 Q-section 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 series-balanced network. The same holds true for the shunt-balanced network, except for what has been written about the simplest form of the matching-stub 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 shunt-balanced, 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 matching-stub 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
driving-point impedance to transmission-line characteristic impedance, this procedure did not sufficiently reduce the standing-wave
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 sending-end 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 **

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 step-up 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 step-down
transformer ratio.

A graphical representation of these effects is given in Fig. 2. Fig. 2-A shows how the resistance and reactance vary
along a piece of 600-ohm line terminated in 300 ohms, and Fig. 2-B shows the variation along a 600-ohm line with a 1200-ohm
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 step-up or step-down, 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.

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 antenna-matching 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.

**Q-Section Transformer **

Fig. 3 shows the popular Q-match, 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 Q-match
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.

**Series-Balanced Network **

In the series-balanced 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)

In the series-balanced 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

**Shunt-Balanced Network **

In the shunt-balanced 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 series-balanced 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

* When a stub is desired at X_{BP}, β is found from (9) or (10).

**Linear Shunt Reactors **

The shunt-balanced 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} - Sending-end 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

**Matching-Stub Network **

In the matching-stub network, which is a special form of the shunt-balanced 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 shunt-balanced 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.

**Example 1 - **

Given: A matching-stub 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 shunt-balanced network with Z_{L} = 8 ohms, Z_{0} = 75 ohms (Twin-Lead), 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
75-ohm Twin-Lead in the transformer section, since the power is high and Z_{L} and Z_{0} are quite different.
To assure a minimum of losses, 1-inch 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 sending-end 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 Q-section 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 purely-resistive 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 self-resonated 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 transmission-line 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 standing-wave ratio is high and the use of
solid-dielectric cable in this section may result in considerable power loss or possibly breakdown. Cables are rated under
flat-line 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 shunt-balanced
network is In the series-balanced 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