Antenna Matching with Line Segments
September 1948 QST

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 graphically trace out what's happening, but you have to admit there's something mystical about it all. Fortunately, Mr. John Marshall published this "Antenna Matching with Line Segments" article in the September 1948 issue of QST magazine.

Antenna Matching with Line Segments

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_T²/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_T²/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 trans-former 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, Z₀, 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 Z₀,

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 Z₀ 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 Z₀) and of such length, θ, that Z_S contains an R_S equal to Z₀ In other words, the segment becomes a transformer having the proper ratio to make Z_L appear equal to Z₀ 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 Z₀.

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 from³:

  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 Z₀, 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 - Z₀) 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 Z₀. The parallel re-actance 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 Z₀.

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 Z₀ 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 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 Handbook¹ 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.

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, Z₀ and Z_C are equal, and design formulas become quite simple. Once a line of known Z₀ has been selected, it is necessary to find only θ and β.

Since this system is of the shunt-balanced type and Z_T = Z₀, tan θ is found from formula (7).

When Z_L < Z₀, an open stub is used and formulas (8) and (9) combine into one operation. Then β may be found from

When Z_L > Z₀, a closed stub is used and for-mulas (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, Z₀ = 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°. Con-verting 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, Z₀ = 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 Z₀ 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 Z₀ 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 Z₀ 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 < Z₀ maximum voltage is at the sending end in any of these networks, while when Z_L > Z₀ 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 < Z₀, and Z_T must be less than

EQUATION HERE

when Z_L > Z₀

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 May 19, 2022
(updated from original post on 4/14/2016)