ANOTHER LOOK AT REFLECTIONS

By: M. Walter Maxwell, W2DU/W8HKK

This series appeared in the April 1973, June 1973, August, 1973, October, 1973, April 1974,December 1974, and the August 1976 issues of QST magazine.

Copyright 1973,1974,1976, 1999 American Radio Relay League. All rights reserved.

NOTE: This series was completed as the book "Reflections", By M. Walter Maxwell, W2DU, Published bythe American Radio Relay League, 1990 (no longer in print).

All references, as well as the bibliography for this series of articles appear at the end of this series.

Compiled with the help of Tom Hammond, NØSS 06/99

This article is the first in a series of QST "Extras" discussing the variousfacets of the operation of rf transmission lines - transmission-line theory, ifyou will. Close attention and study of the material, rather than a casualreading, will be rewarding for those interested in gaining a further knowledgeof the subject. Some background in complex algebra will aid the reader inunderstanding the presentation, although the discussions hinge on simplealgebraic expressions and manipulations of the Smith chart. In later parts ofthe series, the author, M. Walter Maxwell, uses a method of presentationwhich is much more vivid than the conventional way of simply showingvector combinations for a complex load.

Licensed as W8KHK in 1933, Mr. Maxwell's professional antenna experienceincludes participation from 1940 to 1944 in building antenna farms at FCCmonitoring stations in Hawaii and at Allegan, Michigan. With a BS degreefrom Central Michigan University, he has been an engineer with the RCACorporation since 1949, and a charter member of RCA's Astro-ElectronicsDivision. Since 1960 he has been in charge of the antenna laboratory and testrange at RCA's Space Center, Princeton, NJ. More than 30 earth-orbitingspacecraft have antenna systems designed solely by Mr. Maxwell, includingEcho I and all Tiros-ESSA weather satellites. There are many others to whichhe contributed design assistance. He assisted in the design of Apollo's lunar-rover (moon buggy) earth-link antenna, set up its test-range facilities, andperformed many of its pattern, gain, and impedance-matching measurements.He also engineered the prelaunch spacecraft-checkout (ground station)

antenna systems at Cape Kennedy for the Tiros and Relay projects, and had complete engineering responsibility for the rfportion (receivers, transmitters, and antennas) of the five ground-station complexes used in Project SCORE, the orbitingAtlas which broadcast President Eisenhower's "Christmas Message from Space" in December, 1958.

Even though Mr. Maxwell's profession deals with far-out space communications, he is very much a down-to-earth amateur.Now licensed as W2DU, he still holds his original call, W8KHK, as well. In addition, he is the trustee of K2BSA, the stationof the National Headquarters Radio Club, Boy Scouts of America.

Another Look at ReflectionsPart 1 - Too Low a VSWR Can Kill You

BY M. WALTER MAXWELL,* W2DU/W8KHK

JUDGING BY WHAT we hear on the air,nearly everyone is looking for a VSWR of one-to-one. Question why, and the answer may be, "I'mnot getting out on this frequency because my SWRis 2.5:1. There's too much power coming back andnot enough getting into the antenna," or, "If I feed aline having that much SWR, the reflected powerflowing back into the amplifier will burn it up," orstill, "I don't want my feed line to radiate." Any ofthese answers shows misunderstanding ofreflection mechanics, and are symptomatic of thecurrent state of education on this subject. Rationaland creative thinking toward antenna and feed-line

design practice has been absent for a long time,having been replaced with an unscientific andthought-inhibiting attitude, as in the*days beforeCopernicus persuaded the multitudes that theuniverse did not revolve around the earth. Thissituation originated with the introduction of coaxialtransmission lines for amateur use around the timewe got back on the air after World War II, and hasgained momentum since SWR indicators appearedon the scene and since the loading capacitor of the * Engineer. Chief of Space Center Antenna Laboratorand Test Range, Astro-Electronics Division, RCACorporation, Princeton, N.J.

pi-net tank replaced the swinging link as an output-coupling control. We are in this state because somuch misleading information has been, and is stillbeing published concerning behavior of antennaswhich are not self-resonant, feed-line performancein the presence of reflections when mismatched tothe antenna, and especially the meaning andinterpretation of the VSWR data.

Articles containing explicitly erroneousinformation and distorted concepts find their wayinto print, become gospel, and continue to bepropagated with chain-letter effectiveness. Theseinclude such gems of intuitive logic as (1) alwaysrequiring a perfect match between the feed line andantenna; (2) evaluating antenna performance orradiating efficiency only on the basis of feed lineSWR - the lower the better; (3) pruning a dipole toexact resonance at the operating (single) frequencyand feeding with an exact multiple of a half-wavelength coax - no other length will do; (4)adjusting the height - perhaps just lowering theends into an inverted V - to make the resistivecomponent equal to the line impedance; or (5)subtracting percent reflected power from 100 todetermine usable percentage of transmitter outputpower (nomographs have even been published forthis erroneous method). As a result of thesemisdirected concepts, we have been conditioned toavoid any mismatch and reflection like the plague.One-to-one all the way! Sound exaggerated? Not ifthe readers' receivers are tuning the same amateurbands as the author's! In the current vernacular, onecould say we have a severe SWR hangup! In manyinstances, from the viewpoint of good engineering,this hangup is inducing us to concentrate ourimpedance-matching efforts at the wrong end of thetransmission line. (See reference 6.)1

It is ironic that we should be in thissituation, because the amateur is generally quitepractical when it comes to following theoreticalconsiderations. In this case we have been followingthe perfect-match theory down the narrow pathbecause many of the aforementioned articles havemisled us to believe that all reflected power is lost,with never an inkling that, properly controlled,reflections can be turned to our advantage inobtaining increased bandwidth which we arepresently throwing away.

That so much misinformation gainedfoothold is surprising in view of the correctteachings of the ARRL Handbook (ref. 1), the

ARRL Antenna Book (ref. 2), the works ofGrammer (refs. 3, 4, and 5), Goodman (ref. 7),McCoy (refs. 8 through 13 and 41), Drumeller (ref.14), Smith (ref. 15), and especially two articlesaddressed to a subject nearly identical to this one,by Grammer (ref. 6) and Beers (ref 16). Oneobjective of this article, therefore, is to identifysome of the many erroneous notions concerningreflection principles with sufficient clarity tochallenge the reader to question his own positionon the subject. Once we correctly understandmismatch and reflections we can obtainimprovement in operational antenna flexibility,similar to going VFO after being rockbound with asingle crystal. And when we discover how little wegain by achieving a low SWR on the feed line wewill avoid unnecessary and time-consumingantenna modifications, often involving hazardousclimbing and precarious operations on a roof ortower, which can result in injuries or even death.Let's kill SWR misconceptions - not hamoperators!

Open-Wire Versus Coax Feed Lines

The theory behind the transmission ofpower through a feed line with minimum loss byeliminating all reflections - terminating the linewith a perfect match - is equally valid, of course,for open-wire and coaxial lines. But in the days ofopen wire, prior to our widespread use of coax, itwas tempered with practical considerations. Open-wire line was, and still is, used with high VSWR toobtain tremendous antenna bandwidths with highefficiency. This is because all power reflected fromthe line/antenna mismatch which reaches the inputsource is conserved, not dissipated, and is returnedto the antenna by the "antenna tuner" (transmatch)at the line input. But, although the loss fromreflections and high SWR is not zero, thisadditional loss is negligible because of the lowattenuation of open-wire lines. If the line werelossless (zero attenuation) no loss whatever wouldresult because of reflections.

The error in our thinking, that standingwaves on coaxial line must always be completelyeliminated, originated quite naturally, because thepermissible reflection and SWR limits are lowerthan in open wire. When using coax for trulysingle-frequency operation it makes sense to matchthe load and line to the degree economically

feasible. But it makes no sense to match at the loadin many amateur applications where we are chieflyinterested in operating over a band of frequencies;single-frequency operators we are not, except asour misguided concern over increasing SWRrestricts our departure from the antenna resonantfrequency.

Many authors are responsible forperpetuating the unscientific and erroneousviewpoint that the coax-fed antenna must beoperated at its self-resonant frequency, bycontinually overemphasizing the necessity for itsbeing matched to the line within some arbitrary,low SWR value to preserve transmissionefficiency, and by implying that efficiency equals100 minus percent reflected power. The viewpointis unscientific because it neglects the mostimportant factor in the equation for determiningefficiency - line attenuation. And it is alsoerroneous because efficiency does not relate toreflected power by simple subtraction. Setting anSWR limit alone for this purpose is meaningless,because the amount of reflected power actually lostis not dependent on SWR alone. The attenuationfactor for the specific feed line must also beincluded because the only reflected power lost isthe amount dissipated in the line because ofattenuation - the remainder returns to the load.These authors have so wrongly conditioned usconcerning what happens to the reflected powerthat many of us have overlooked the correctapproach to the subject. It is clearly presented inboth the ARRL Handbook and the Antenna Bookthat transmission efficiency is a two-variablefunction of both mismatch and line attenuation.With this knowledge and by using a graph of thefunction appearing in these ARRL books,presented here as Fig. 1, the amateur can determinehow much efficiency he will lose for a given SWRwith the attenuation factor of each specific feedline. He can then decide for himself what therealistic SWR limit should be.

Unimportance of Low SWR Values

In our efforts to obtain low feed-line SWRsof 1.1, 1.2, or even 1.5 to 1, we have gone far pastthe diminishing-returns point with respect toefficient power transfer, even for single-frequencyoperation, for the same reason one would notinstall a No. 4 or 6 wire in a house wiring run

where No. 12 is sufficient. Reference to the basictransmission-line equations, which have alwaysbeen readily available in engineering texts andhandbooks (refs. 1, 2, 17, 18, 19, 33), will verifythis analogy in addition to making it clearlyapparent that authors who simply insist on lowSWR or find 1.5 or 2 to 1 objectionably high havefailed to comprehend the true relationship betweenreflected and dissipated power. From the viewpointof amateur communications, it can be shownmathematically and easily verified in practice thatthe difference in power transferred through anycoaxial line with an SWR of 2 to 1 is imperceptiblecompared to having a perfectly matched 1.0-to-1termination, no matter what the length orattenuation of the line, and that many typicalcoaxial feed lines that we use in the hf bands withan SWR of 3 or 4, and often as high as 5 to 1, havean equally imperceptible difference. When feed-line attenuation is low, allowing such higher valuesof SWR permits operating over reasonably widefrequency excursions from the self-resonantfrequency of the antenna with the imperceptiblepower loss just described, in spite of the popularimpression to the contrary.

The relative unimportance of low SWRwhen feed-line attenuation is low is demonstratedrather vividly in the following two examples ofspacecraft antenna applications. First, in the Tiros-ESSA-Itos-APT weather satellites, of which theentire multifrequency antenna-systems design wasthe work of the author, the dipole terminalimpedance at the beacon-telemetry frequency (108MHz in early models) was 150 - j100 ohms, for aVSWR of 4.4, reflected power 40 percent.Matching was performed at the line input, where itwas fed by a 30 milliwatt telemetry transmitter.(We can't afford much power loss here!) The feed-line and matching-network attenuation was 0.2 dB,and the additional loss from SWR on the feed linewas 0.24 dB (5.4 percent), for a total loss of 0.44dB (9.6 percent). On the prevalent but erroneousassumption that all reflected power (40 percent) islost, only 18.1 milliwatts would reach the antenna,and efficiency, determined on the same erroneousbasis, would be only 60 percent. But 27.1milliwatts were measured; of the 2.9 milliwatts lostin total attenuation, only 1.6 milliwatts of it wasfrom the 4.4:1 VSWR. So the real efficiency wouldhave been 95.5 percent if perfectly matched at theload, but reduces to 90.4 percent by allowing the

4.4 VSWR to remain on the feed line. Second, inthe Navy Navigational Satellite (NAVSAT), usedfor precise position indications for ships at sea, theantenna terminal impedance at 150 MHz is 10.5 -j48 ohms, for a VSWR of 9.8, reflected power 66percent. Also matched at the line input, flat-lineattenuation is 0.25 dB, and the additional loss fromSWR is 0.9 dB, for a total system loss of 1.15 dB,approximately 1/6 of an S unit. This is aninsignificant amount of loss for this situation, evenin a space environment where power is at apremium. Why did we match at the line input?Because critical interrelated electrical, mechanicaland thermal design problems made it impractical tomatch at the load. Line-input matching provided asimple solution by permitting the matchingelements to be moved to a noncritical location.This design freedom afforded tremendous saving inengineering effort with negligible compromise in rfefficiency, in spite of SWR levels many amateurs

would consider unthinkable.Another factor which contributes to

misunder-standing is the confusion between twodistinct, line-usage conditions - one of constantincident voltage, and the other of constant inputpower (for relative amplitudes, see ref. 19, Fig. 1.3,page 6, and Fig. 3.6, page 29). Laboratory andexperimental work often requires holding incidentvoltage constant with variation in loading. Aconstant-voltage source is usually obtained for thispurpose by inserting a pad having 15 to 20 dBattenuation between the generator and the line toabsorb the reflected power, preventing it fromreaching the generator where it would alter the linecoupling and cause the generator output voltage tovary. Because of the absorption of the pad, thegenerator sees a nearly perfect match for all loadconditions and all reflected power is lost - but theseare laboratory control conditions required to obtainvalid test data.

When we amateurs make a change thatalters line loading, which in turn alters thetransmitter-to-line coupling because of returningreflected power, we can readjust the coupling,returning the line-input power (not forward power)to its previous value regardless of the reflectedpower value.2 We amateurs can adjust coupling forchanges in loading - in the laboratory this isnot convenient. We amateurs use low-attenuationlines to conserve reflected power - laboratorysetups insert attenuation to dissipate it. Confusionover these distinctions has helped perpetuate theerroneous "lost reflected power" concept.

As a result of these variousmisunderstandings, many amateurs never evenwonder whether there are any benefits to be gainedby not matching at the line-antenna junction. Manynow even shun the use of open-wire lines (not theOld Timers), completely missing the joy of a QSYto the opposite end of the band with only a simplechange in transmatch tuning, because the fear ofreflections engendered by the exaggeratedapplication of the theory to coax has crept into thethinking concerning any form of mismatchedconnection. Adding still further to the confusion isthe old-wives' tale that the reflected power isdissipated in the transmitter, causing tube and tank-coil heating and all kinds of other damage. Thismyth developed out of ignorance of the truemechanics of reflections and became the easy, butfallacious, explanation of what seems to be

abnormal behavior in the transmitter when feedinga line with reflections. What really happens at thetransmitter is simply a change in coupling, whichwill be explained in detail in a section to follow.Then we may understand how to operate withabsolutely no danger of damaging the amplifierwhile feeding into a line with high SWR.3

Engineering an Amateur Antenna System

Engineering is the process of makingworkable compromises in design goals wheretheories guiding different aspects of the design arein conflict, making it impossible to optimize all thegoals. Good engineering is simply recognizing thecorrect choices in the compromises and relaxingthe right goals, as in the spacecraft antenna designmentioned earlier. We amateurs spend many hoursbudding and pruning antenna systems. Wouldn't itbe worthwhile spending some of that time learninghow to engineer the design in order to make correcttrade-off decisions among related factors instead ofletting King VSWR dictate the design?

FIRST, we need to improve our knowledgeof reflection mechanics and transmission-linepropagation to understand…

1) why reflected power by itself is anunimportant factor in determining how efficientlypower is being delivered to the antenna.

2) the effect of line attenuation (to discoverwhy it is the KEY factor which will tell us whenand how much to be concerned with reflectedpower and when to ignore it).

3) why all power fed into the line, minusthe amount lost in line attenuation, is absorbed inthe load regardless of the mismatch at the antennaterminals.

4) why reflection loss (mismatch loss) iscanceled at the line input by reflection gain (ref.19, p. 36, and ref. 25, part 2, p. 33).

5) why a low SWR reading by itself is nomore a guarantee that power is being radiatedefficiently than a high SWR reading guarantees it isbeing wasted.

6) why SWR is not the culprit in transmitterloading problems - why the real culprit is thechange in line-input impedance resulting from theSWR, and why we have complete control over theimpedance without necessarily being concernedwith the SWR.

7) the importance of thinking in terms ofresistive and reactive components of impedanceinstead of SWR alone, and why SWR by itself isambiguous, especially from the viewpoint of theselection and adjustment of coupling and matchingcircuitry.

SECOND, we need to become aware thatwith moderate lengths of low-loss coax, such as wecommonly use for feed lines, loss of power becauseof reflected power in the hf bands can beinsignificant, no matter how high the SWR. Forexample, if the line SWR is 3, 4, or even 5 to 1 andthe attenuation is low enough to ignore thereflected power, reducing the SWR will yield nosignificant improvement in radiated power becauseall the power being fed into the line is alreadybeing absorbed in the load. This point has especialsignificance for center-loaded mobile whips,because of the extremely low attenuation of theshort feed line.

THIRD, we should become more familiarwith the universally known, predictable behavior ofoff-resonance antenna-terminal impedance and itscorrelation with SWR. This knowledge provides ascientific basis for evaluating SWR-indicatorreadings in determining whether the behavior ofour system is normal or abnormal, instead ofblindly accepting low SWR as good, or rejectinghigh SWR as bad. The following two examplesemphasize the importance of this point by showinghow easily one may be misled by a low SWRreading:

1) A ground system having 100 properlyinstalled radials has negligible loss resistance (ref.20). Many a-m broadcast stations use 240 radials,while the FCC requires a minimum of 120. Withsuch a ground system the terminal impedance ofthe average quarter-wave vertical is 36.5 + j22ohms, and approximately 32 ohms when shortenedto resonance. When fed with a 50-ohm line, theSWR at resonance will be close to 1.6, risingpredictably on either side of resonance. But a 15-radial ground system will have approximately 16ohms of ground-loss resistance with this antenna. Ifwe remove a few radials at a time from the 100-radial dial system, the increasing ground resistance,added to the radiation resistance, increases the totalline-terminating resistance. The terminatingresistance comes closer and closer to 50 ohms,reducing the SWR. When enough radials have beenremoved for the loss resistance to reach 18 ohms,

the terminating resistance will be 50 ohms for aperfect one-to-one match! But while the SWR wentdown, so did the radiated power, because now thepower is dividing between 32 ohms of radiationresistance and 18 ohms of ground resistance!

Ground systems having from two to fourradials may have a loss resistance as high as 30 to36 ohms, so now the resonant-frequency SWR willbe around 1.4 or 1.5. But instead of rising from thisvalue, as it should at frequencies away fromresonance, the ground loss holds the off-resonantSWR to low values. The low SWR simplyindicates that the line is well matched, but it offersno clue that approximately half the power isheating the ground.

2) Some amateurs who employ a one-to-one balun believe that "one-to-one" means it willprovide a one-to-one match between the feed lineand the antenna. This is a serious error because"one-to-one" only specifies the output-to-inputimpedance ratio - no matter what impedanceterminates the output, the same value is seen at theinput. Nevertheless, these amateurs are convincedthe baluns are "matching," because the SWRsometimes goes down dramatically when the balunis inserted. Often with a balun the SWR is less than2:1 over the entire 75-80 meter band, wheresomewhat over 5:1 is normal at the band ends.

Off-resonance SWR is reduced herebecause the ferrite core of the balun saturates whileattempting to handle the reactive current, whichnow exceeds the maximum core-current level.Thus, the full excursion of the reactive componentof antenna impedance is prevented from appearingat the balun input. All power above the saturationlevel is lost in heating the balun, while the lowSWR is deceiving the unsuspecting amateur.

The true SWR will be unchanged by a 1:1balun with a core capable of handling the currentwithout saturating (if it has no significant leakagereactance).4 However, the SWR indicator may notshow the true SWR without the balun if antennacurrent on the outside of the coax is present at theSWR meter (ref. 36).

So it is important to know approximatelywhat SWR to expect - if it is low, determinewhether it should be. Don't assume that a low SWRindicates success, or guarantees a great system! Beespecially suspicious if the SWR remains low orrelatively constant over a moderate frequencyrange, unless specific broad- banding steps have

been performed on the radiating system. Thisknowledge is elementary and routine for theantenna design engineer, but too little informationin this area has been available for the amateur,considering the degree of his involvement withantennas. While antenna-terminal impedancebehavior with frequency is shown in the ARRLAntenna Book (ref. 2, Fig. 2-7), correlation of theimpedance change with SWR will be covered indetail later, to enable us to predict normal SWR,within limits, with a nonresonant antennaterminating the feed line.

FOURTH, we need to reexamine the use ofopen-wire lines as tuned lines (refs. 3, part 3, p. 20;10; and 21, p. 23), to discover that the principlesused there are exactly what we have beendiscussing. Remember, with tuned lines wecompletely ignore the mismatch at the antenna end,and compensate for the mismatch with the tuner atthe input end, over the entire frequency range ofthe band. The SWR may run as high as 10, 15, oreven 20 to 1, but the power reflected from themismatch is re-reflected back to the antenna by thetuner. Tuning for maximum line current simplyadjusts the phase of the reflected wave to rereflectdown the line in phase with the forward wave,again reaching the antenna. Thus the reflection lossfrom the mismatch is canceled by the reflectiongain of the tuner.

Many of us amateurs know from age-oldpractice that a 600-ohm line made of two No. 12wires on six-inch spacing would work every time.We had little incentive to learn how they worked -why they transferred power efficiently with suchhigh reflected power and SWR, or that adjustingthe reflected-wave phase to rereflect in phase withthe forward wave was just another way of viewingthe reactance cancellation required to obtainmaximum line and antenna current. Hence, ourmisunderstanding of the similarity between open-wire and coaxial-line operation with mismatchedloads. The principle is the same in both, only thedegree is different. In other words, for manyapplications, coax can be used as a tuned line inprecisely the same manner as open wire. Thespacecraft systems mentioned earlier are examples.

Thus, coax connected directly into theantenna may be operated with substantialmismatch. In this case, the SWR limits whileoperating away from the self-resonant frequency ofthe radiator are determined entirely by power lost

because of line attenuation. Voltage breakdown andcurrent heating should not be a problem at our legalpower limit with RG-8 or -11/U, or with RG-58 or-59/U at lower powers. because voltage at an SWRmaximum is only SWR times the matched value.The line-input impedance will no longer be 50ohms, but depending on the magnitude of themismatch and length of the cable, we maydetermine whether the output tank of thetransmitter has sufficient impedance-matchingrange (surprisingly high in some rigs, none inothers) to permit feeding the line directly (ref. 4,part III), or whether an intermediate matchingdevice (transmatch, or other type of tuner (refs. 9-12, incl., 22) will be required to adjust for correctcoupling between the line and the transmitter.(Balun and filter use will be discussed later.) Theimportant point we are emphasizing is that, withinthe limits mentioned, all required matching may betransferred back to the operating position insteadof forcing the match to occur at the antenna feedpoint - without suffering any SIGNIFICANT lossin radiated power. The use of this technique, whichmay come as a surprise to many, does notcontradict any theory. It is actually an embodimentof the fundamental principle of network theory

called conjugate matching, (refs. 17, p. 243; 19, p.38; 35, p. 49) which is the basis for all antennatuner, or transmatch, operation with either open-wire or coaxial lines.

After learning of the benefits obtained withline-input, or conjugate matching in the twospacecraft examples described earlier, it isinteresting to compare the results using this sameinput matching technique in typical 80- and 40-meter situations. Eighty meters is the widestamateur band in terms of percent of centerfrequency and thus suffers the greatest SWRincrease with frequency excursion to the band ends.A dipole cut for resonance at 3.75 MHz will yieldan SWR in a 50-ohm feed line somewhat above 5:1at both 3.5 and 4.0 MHz. As shown in Fig. 2, in a100-foot length of nonfoam RG-8/U, an SWR of5:1 adds only 0.46 dB loss to the matched (i.e., flatline) loss of 0.32 dB at 4.0 MHz.. So out almost tothe band ends, less than 1/12 of an S unit is lostbecause of the SWR, an imperceptible amount.This further verifies the principle and proves thatfull-band, coax-fed dipole operation on 80 metersalso is practical. Even with the high SWR at theband ends, the loss cannot be distinguished fromwhat it would have been had the SWR been aperfect one-to-one! At 40 meters, with the dipoleresonated at 7.15 MHz, something is amiss if theSWR exceeds 2.5 at the band ends. And from Fig.2 it may be seen that this SWR adds only 0.18 dBto the matched loss, which at 7 MHz is 0.44 dB for100 feet of RG-8/U coax.

Nonreflective Load Versus Conjugate Matching

Now is perhaps a good time for the readerto contemplate the conflict between the no-reflection perfectly matched load theory and theconjugate match theory. It is amply evident fromthe standpoint of good engineering that as long asSWR does not exceed the value above which onecannot afford to compromise further power inexchange for improved operating flexibility, theconvenience and increased bandwidth afforded byconjugate matching at the line input is obvious.

But it also presents a real challenge tolearning more about complex impedance, becausethe line-input impedance now has resistive andreactive components, both of which vary withchanges in line length and with frequency in thepresence of reflections. Thus, we need to

understand complex impedance in order to chooseand adjust correct conjugate matching circuitry tocouple the transmitter to the line, or to adjust thetransmitter directly to the line if sufficientmatching range is available. Practically allproblems encountered while attempting to obtainproper coupling or loading to a line with reflectionscan be traced simply to not understanding thecorrelation of line length and relative phase of theincident and reflected waves with the resultingcomplex impedance seen at the input terminals ofthe line.

A detailed discussion of reflectionmechanics and feed-line propagation will be

presented in subsequent installments. Included willbe a novel means for explaining impedancetransformation along the line in direct relation toincident and reflected waves, which will simplifythe understanding of what does and does nothappen when a line length is changed, and how toselect the correct length for given conditions. Therelation of line attenuation to permissible SWRwhile using conjugate matching techniques, alongwith details on how to obtain proper coupling andloading of a transmitter to a line for which the inputimpedance has changed because of reflections, willalso be presented.

Part 2 - Countdown for a JourneyFrom Mythology to Reality

Part 1 of this series of articles appeared inQST for April, 1973. In that part we saw thatobtaining a low SWR is relatively unimportant foran efficient transfer of power when line attenuationis low. Four steps to assist in understanding theoperation of lines with reflection were suggested,and the concept of matching the compleximpedance at the line input in the presence ofreflections, called conjugate matching, wasintroduced. The paragraphs which follow presentfor consideration some of the basic principlesinvolving efficient power transfer through any lineterminated in a mismatch.

A conjugate match exists throughout theentire system when the internal resistance of thesource is made equal to the resistive component ofthe line-input impedance (or vice versa) and allresidual reactance components in the source andline-input impedances are canceled to zero. In thiscondition the system is resonant. All availablepower from the source enters the line, andreflections from any terminating mismatch or otherline discontinuities are compensated by acomplementary reflection obtained by introducinga nondissipative mismatch at the conjugate matchpoint. This nondissipative mismatch is one which ifplaced in the system by itself, would produce thesame magnitude of reflection, or SWR, as isproduced by the mismatched line termination. Theresult is a precise and total rereflection of thearriving reflected wave. Andrew Afford makes amagnificent presentation of this concept (ref. 39,pages 10-15). Although it sounds verycomplicated, this entire set of conditions isautomatically fulfilled simply by completing acorrect tuning and loading procedure. It matters notwhether a transmitter having sufficient matchingrange feeds the line directly, or whether an externaltransmatch is used where additional range isrequired. If the source generator is now replaced bya passive impedance equal to its internalimpedance5 the line can be opened at any point.And looking in either direction, one will see theconjugate of the impedance seen in the opposite

direction - whatever R + jX value is seen in onedirection, R - jX is seen in the other.

Contrary to our prevalent, deeply ingrainedbelief, it is therefore not true that when atransmitter delivers power into a line withreflections, a returning reflected wave always seesthe internal generator impedance as a dissipativeload and is converted to heat and lost. It canhappen under certain conditions of pulse-typetransmission; for instance, if the generator is turnedoff after delivering a single pulse into the linewhile retaining its internal impedance across theline, the returning pulse wave will be absorbed. Butif a conjugate-matched generator is activelysupplying, power when the reflected wave returns,the reflected wave encounters total reflection at theconjugate match point and is entirely conserved,because it never sees the generator resistance as adissipative terminating load. This is because thesource and reflected voltages and currentssuperpose, or add at the match point, just as if thereflected power had been supplied by a separategenerator in series with the source. And since thesource voltage is generally greater than thereflected, the sum of their voltages yields a netcurrent flow which is always in the forwarddirection.6 The reflected power adds to the sourcepower, deriving reflection gain which compensatesfor the reflection loss suffered at the mismatchedtermination.

Line Losses

All reflected power reaching the source isreturned to the load, as part of the forward orincident wave. The only reflected power lost isbecause of line attenuation, during its return to thesource and once again during its return to the load.The higher the line attenuation, the less reflectedpower reaches the source to add to the forwardpower. Thus, the lower the line attenuation, thehigher the allowable SWR for a given loss becauseof SWR. No reflected power is lost in a losslessline, no matter how high the SWR, because it allgets ultimately to the load. This is why open-wire

line functions efficiently as a tuned line with anyreasonable mismatch value -- its attenuation isalmost negligible. Attenuation (being higher incoax) imposes lower limits on the mismatch andmay require calculation of the loss penalty for agiven SWR. Both the attenuation and SWR must bequite high to incur any substantial additional lossover and above the matched-line loss.7

Coax has higher rf losses than open wire athf chiefly because of its lower impedance, causinghigher current flow at lower voltage for the samepower. This results in higher I2R loss for the sameeffective conductor size. (Electric powerdistribution lines minimize I2R loss by use of highvoltage and low current). Skin effect increases theloss with rising frequency because of decreasedeffective conductor size, but only at VHF andhigher does the dielectric loss become a substantialcontributor to the attenuation factor. From this it isunderstandable why RG-8/U, especially the foamtype with its larger center conductor (ref. 23), willallow higher SWR (more bandwidth) than RG-58/U for the same additional loss penalty. And forany cable, the shorter it is the less loss is added fora given SWR.

A fifth step in improving our understandingof the reflected-power problem is to view thesituation objectively, asking yourself, "Have Ifallen prey to any of the erroneous teachings? Can Ispot the wrong dope when I hear it discussed? Do Iunderstand the principles well enough to convinceothers of the correct version if the opportunityarises?" Several pertinent short statements followwhich may be used as self-test material. They high-light and summarize many reflection-relatedconcepts known to be generally confused amongthe amateurs. All of the statements are TRUE. Inthe interest of brevity they are not intended to becompletely self-explanatory, but sufficient materialfor obtaining a complete understanding of eachpoint will appear in later installments, or isavailable in bibliography references included inpart 1. Support for nearly every statement can befound in The ARRL Antenna Book alone.

True or False?

1) Reflected power does not represent lostpower except for an increase in line attenuationover the matched-line attenuation. In a losslessline, no power is lost because of reflection. Only

when the flat-line attenuation and SWR are bothhigh is there significant power lost from reflection.On all hf bands with low-loss cable, reflectedpower loss is generally insignificant, though atVHF it becomes significant, and at UHF it is ofextreme importance.

2) Reflected power does not flow back intothe transmitter and cause dissipation and otherdamage. Damage blamed on reflections is reallycaused by improper output-coupling adjustment –not by SWR. Tube overheating is caused by eitheror both overcoupling and reactive (mistuned)loading. Tank-coil heating and arc-overs resultfrom a rise in loaded Q caused by undercoupling.With some manipulation, proper output coupling(indicated by a normal resonant plate-current dip atthe correct loading level) can be attained no matterhow high the SWR. The transmitter doesn't "see"an SWR at all -- only an impedance resulting fromthe SWR. And the impedances are matchablewithout concern for the SWR. This is one of themost important points of confusion at issue.

3) Any effort to reduce an SWR of 2:1 onany coaxial line will be completely wasted from thestandpoint of increasing power transfersignificantly. (See Fig. 1, part 1.)

4) Low SWR is not proof of a good-qualityantenna system or that it is working efficiently. Onthe contrary, lower than normal SWR exhibitedover a frequency range by a straight dipole or avertical over ground is a clue to trouble in the formof undesired loss resistance. Such resistance can befrom poor connections, poor ground system, lossycable, and so forth.

5) The radiator of an antenna system neednot be of self-resonant length for maximumresonant current flow, the feed line need not be ofany particular length, and a substantial mismatch atthe line-antenna junction will not prevent theradiator from absorbing all real power available atthe junction. (refs. 3, part 3, p. 10; 24.)

6) If a suitable transmatch cancels all thereactance developed by a nonresonant-lengthradiator and a random-length feed line which ismismatched at the antenna feedpoint, the antennasystem is resonant, the mismatch effect is canceled,maximum current flows in the radiator, and all thereal power available at the feed point is absorbedby the radiator.

7) The majority of tower radiators used inthe standard a-m broadcast band (from 540 to 1600

kHz) are of heights which are not resonant lengthsat the frequency of operation.

8) SWR on the line between the antennaand transmatch is determined only by the matchingconditions at the load, and is not changed or"brought down" by the matching device. "LowSWR" obtained by using the device indicates onlythe mismatch remaining between the inputimpedance of the transmatch and impedance of theline from the transmitter.

9) Adjusting the transmatch for maximumline current creates a perfect mirror termination forthe reflected wave, causing it to be totallyrereflected on arrival at the input. The tunerprovides the proper reactance to cancel the equalbut opposite reactance resulting from the amplitudeand phase difference between the source andreflected waves at the input. This causes thereflected wave to add in phase to the source waveto derive the incident power, which is the sum ofthe source and reflected power.

10) Total rereflection of the reflected powerat the line input is the reason for its not beingdissipated in the transmitter, and why it isconserved, rather than lost.

11) With a good "antenna tuner" ortransmatch and a well-constructed open-wire

feeder, a 130-foot center-fed dipole will not radiatesignificantly more power on 80 meters than one 80feet long for the same power fed from thetransmitter (refs. 10; 21; 3, part 3, p. 20: 7, pp. 50and 126).

12) A dipole cut to be self-resonant at 3.75MHz and fed with either RG-8/U or RG-11/U coaxwill not radiate significantly more on 3.75 MHzthan on 3.5 or 4.0 MHz with any feeder length upto 150 or 200 feet.

13) With the 3.75 MHz dipole the feed-lineSWR will rise to around 5.0 at both 3.5 and 4.0MHz, thus utilizing the coax as a tuned feeder, butwith insignificant loss in radiated power across theentire 80-meter band.

14) With the use of a transmatch or asimple L-network at the line input, proper couplingbetween the transmitter and the tuned-coax feedercan be attained over the entire band with anyrandom coax length.

15) From the standpoint of line loss becauseof SWR resulting from the change in quality, of theimpedance match between the line and antenna,changing the height of the dipole above ground orlowering the ends of a horizontal dipole to make aninverted-V will have an insignificant effect on theamount of power reaching it from the transmitter.

16) As a tuned line at 4.0 MHz, RG-8/Uwill handle 700 watts CW continuously, withinratings, at an SWR of 5:1. With the duty cycle ofSSB it is far below maximum ratings at 2 kW PEP.With a 100-foot length, the total attenuation (SWR= 5) is just 0.8 dB (0.46 dB because of SWR),which is insignificant in terms of received signalstrength.

17) If a line length is critical in order tosatisfy a particular matching condition, the sameinput impedance can be obtained with any length ofline, shorter or longer, by adding a simple Lnetwork of only two components: either twocapacitors, two inductors, or one of each,determined by the specific impedance changerequired of it. This statement is pertinent to coiled-up coax in mobiles. (Refs. 19, pp. 118-128: 24; 30,p. 48; 3 1. )

18) High SWR in a coaxial transmissionline caused by a severe mismatch will not produceantenna currents on the line, nor cause the line toradiate (refs. 32; 2, p. 101).

19) High SWR in an open-wire line at hfcaused by a severe mismatch will not produce

antenna currents on the line, nor cause the line toradiate, if the feed currents in each wire arebalanced, and if the spacing is small at thewavelength of operation (also true at VHF if sharpbends are avoided.) (ref 2, pp. 101, 106.)

20) Both coax and open-wire feed linesmay radiate (ref. 32), though not at a significantlevel, by re-radiating energy coupled into the linefrom the antenna because of asymmetricalpositioning with respect to the antenna. Thecoupled energy results in antenna currents flowingon the outside of the outer coax conductor, or in-phase currents flowing in the wires of the open-wire line. But this condition has no relation to levelof the line SWR in either case (ref. 2, pp. 101,106).

21) SWR indicators need not be placed atthe feed-line/antenna junction to obtain a moreaccurate measurement. Within its own accuracylimits, the indicator reads the SWR wherever it islocated in the line. The SWR at any other point onthe line may be determined by a simple calculationinvolving only the SWR at the point ofmeasurement, the line attenuation per unit length(available in a later installment), and the distancefrom the measured point to the point where theSWR is desired.

22) SWR in a feed line cannot be adjustedor controlled in any practical manner by varyingthe line length (ref. 7, p. 51).

23) If SWR readings change significantlywhen moving the bridge a few feet one way or theother in the line, it probably indicates "antenna"current flowing on the outside of the coax, or elsean unreliable instrument, or both, but it is notbecause the SWR is varying with line length. Somewriters insist the bridge must be placed at a half-wave interval from the load to obtain a correctreading. This is incorrect. All readings are invalidif they change significantly along the line, eventhough they may repeat at half-wavelengthintervals (ref. 2, pp. 101, 106, and 132).

24) Any reactance added to an alreadyresonant (resistive) load of any value for thepurpose of compensation to reduce the reflectionon the line feeding the load will, instead, onlyincrease or worsen the reflection. It is for thisreason, though contrary to the teaching of severalwriters, that lowest feed-line SWR occurs at theself-resonant frequency of the radiating element itfeeds, completely independent of feed-line length.

Any measurements which contradict this indicatethat either the measuring equipment or thetechnique (or both) are in error.

25) Of the several types of dipoles, such asthe thin wire, folded, fan, sleeve, trap, or coaxial,none will radiate more field than another, providingeach has insignificant ohmic losses and is fed thesame amount of power (ref. 3, part 3).

26) If coax at least the size of RG-8/U isused in mobile installations (80 thru 10 meters),any matching required to load the transmitter maybe done at the cable input end without significantpower loss compared to matching at the antennaterminals, and with improvement in operatingbandwidth.

27) With center-loaded mobile whips ofequal size having no matching arrangement at theinput terminals, best radiating efficiency isobtained on models having the lowest measuredterminal resistance (highest resonant SWR, modelfor model). Models having lowest SWR arewasting power in the loading coil, because of eithera low value of coil Q or excessive distributed coilcapacitance, or both.

As was mentioned earlier, all of thesestatements are true. These examples have beencentered around 80-meter operation becausebandwidth and dipole length on this band presentthe maximum SWR problem. Of all the amateurbands, 80M has the largest bandwidth, 13.3 percentof center frequency, as compared to 4.2 percent on40M, 2.5 percent on 20M, 2.1 percent on 15M, and5.9 percent on 10M. Having the longestwavelength (excepting 160, of course), 80M posesthe greatest problem with respect to physicalconstruction of radiating systems on existing realestate. Some properties just won't permit an entirehalf wavelength on 80. So these examples shouldhave a special interest for the fellow who wishes towork 80 meters but is forced to use a short antenna.Since the bandwidth and antenna-length problemare really one and the same, the 80-meter exampleshave maximum practical value. But regardless ofwhich band we select, the principles are the same.Practices recommended at the 80-meter level arealso valid on the higher hf bands. Interestinglyenough, as we go to the higher bands, where theline losses increase, the percentage bandwidth ofthe amateur band decreases. This means inherentlylower maximum SWR values will be obtained

during frequency excursions from the design centerto the band ends.

About This Series of Articles

The original idea for writing this paper wasborn while the writer was listening to andparticipating in many discussions concerningmismatch and reflections. It soon became apparentthat if the twenty-seven true statements above werepresented as a true-false test, many amateurs wouldmark them false. Those discussions also revealedthat many experiments were performed in this areafor which the results were predestined to futility,because the experimenter misunderstood theprinciples. In many cases, the experimenter wasunaware of both the ultimate futility of his effortsand that the futility was a direct result of hismisunderstanding. Then the appearance of thearticle by Drumeller (ref. 14) motivated the desireto do something constructive about the problem.Although the results of his experiment agree withthe writer's teachings, Drumeller's preface impliesthat little is known on the subject of mismatch.This is not true, unless he was referring specificallyto us amateurs.

The limited equipment available foramateurs to make precise and completemeasurements at rf, as compared to DC, low-frequency AC, and audio, understandably limits thequality of our experiments. But this doesn't excuseus from needing to know some of the fundamentalprinciples of transmission lines -- it actually makesknowledge of the fundamentals all the morenecessary, so we can correctly diagnose andlogically evaluate the limited measurement data wedo obtain. Our experimental efforts can becomemore productive if we are able to visualize anaccurate physical picture of the voltages, currents,and fields, and how they interact on the line.

So the writer initially planned to compileand publish an extensive bibliography of accuratereferences on the subject, references which arereadily available to the amateur. But at one of themonthly Colts Neck meetings, a good friend andfellow 3999er, the late John Marsh, W3ZF,convinced the writer that a mere list of referenceswould receive little attention. W3ZF thenproceeded to fire real enthusiasm for writing anextensive paper which would not only complementand unify the references, but would also underscore

the problem and stimulate interest in studying thesubject. He felt strongly that this approach wouldmake a more worthwhile contribution to amateurradio.

John's suggestion was followed, and as thework progressed it became apparent that additionalreferences from professional engineering sourceswould be valuable in providing access to greaterdepth of study for the more advanced amateur, andby affording reliable sources for verification ofpoints which have been clouded by controversyamong amateurs.

During study of the numerous referencesfor preparation of the manuscript, an interestingand valuable aspect emerged: the ARRL AntennaBook already unifies the other references, plusmore, because in one handy volume it containssufficient, well-presented material on every pointof importance to the amateur's needs concerningantennas and transmission lines, including thefundamental principles! For the amateur who hasno access to the other references, but could manageto acquire just one book, the Antenna Book is it!Every amateur who feeds rf into an antenna shouldhave a copy -- and read it! One would be surprisedto learn how many antenna engineers have a copyin their own personal libraries.

With the exception of the ARRL AntennaBook, the details of the reflection activity in alength of line seem to be somewhat obscure in theamateur literature, so the beginning of the next partof this series presents a simplified treatment of theinteraction of the fields, currents, and voltages in aline. Following that, a new vector-type presentationusing the Smith chart is given.8 The presentationenhances the physical picture of standing-wavedevelopment, input impedance, and other activityalong the line, and forms a basis for an arithmeticalproof that the explanation presented is correct. Thetwo-generator concept (ref. 17, p. 133) is used asthe basis for determining line-input impedance,because this concept permits direct comparison ofthe incident -- and reflected-wave vectorpresentation with simple series-equivalent circuits.This reduces the line reflection problem to one ofsimple Ohm's Law calculations, and at the sametime achieves a built-in proof of validity. Thisconcept is simple to grasp, easy to remember, andwill be recognized as a powerful tool for assistingthe amateur in analyzing any feed-line situationfrom the viewpoint of matching the transmitter to

the line input under any SWR condition. It usessimple arithmetic, but it is not short and it willrequire some study -- there isn't any short cut tounderstanding reflections on transmission lines.

For those who are unfamiliar with theSmith chart it is easy to learn (ref. 25, part 1). Thechart will be found to be a valuable tool. It hasseparate sets of circles for resistance and reactance.To plot a single complex impedance R + jX onesimply finds the point where the appropriate R andjX circles intersect. Knowledge of the Smith chartwill be found helpful in understanding and devisingmatching circuitry; it is actually fun to use for this

purpose. Additional references of interest areincluded in the bibliography.

The author wishes to express appreciationto his many friends who offered suggestions andcriticisms, especially to Bob Allen, W8IO; KenMacLean, W2KKM, the author's engineeringsupervisor for many years; and colleague BertSheffield, W2ANA. Their help has beeninvaluable. Acknowledgment with thanks is alsogiven to Mr. Phillip H. Smith, ex-1ANB, for hiskind permission to use the Smith chart in theStudies which follow.

Part 3 - Going Around in Circles to Get to the Point; Basic Reflection Mechanics

Basic Reflection Mechanics

It is generally well understood that the sizeor magnitude of a reflection arising from amismatched line termination is determined by thedegree of the mismatch, or how much of theincident-wave power is unabsorbed by the load,expressed as a voltage or current ratio relative tothe size of the incident wave.9 The reflectioncoefficient, ρ (rho),10 is determined quantitativelyfrom the line and load impedances by theexpression

ρ =−+

Z ZZ Z

L C

L C (Eq. 1)

where ZL is the complex load impedance, R +jX,and ZC is the characteristic impedance of thetransmission line. This shows immediately that ρ =0 (no reflection) when ZL = ZC. However, the loadmust be purely resistive (R + j0) for zero reflection,because we are considering only lossless and low-loss lines with a characteristic impedance this ispurely resistive. Perhaps somewhat less appreciatedthan reflection magnitude is reflection phase,which is determined by the character of themismatch, and expressed as an angle relative to thephase of the incident wave. The magnitude ratio, ρand the relative phase, θ (theta.), together comprisethe complex reflection coefficient ρ = ρ∠ θ whichtells us all we need to know about the reflection inorder to use it in understanding transmission-linepropagation and matching techniques. How we useit will be explained in a later section.

An open-circuit (infinite impedance), ashort-circuit (zero impedance), or a purely reactiveload on a transmission line is incapable of

[EDITOR'S NOTE: Here the author is notreferring to the standing wave ratio whichmight be measured with an ordinary SWRindicator. Instead he is referring to travelingwaves. The difference is discussed again in latertext.]

absorbing any power from an incident wave andwill therefore cause total reflection of both volt andcurrent incident waves. The reflection coefficientmagnitude, ρ, at such a load is therefore unity (1.0)for both voltage and current. The step-by-step bywhich the reflections arise on low-loss lines, bothcoaxial and open wire, is as follows. The incidentelectromagnetic wave sees the characteristicimpedance, ZC, of the line as a resistive load as itleaves the generator in its forward travel down theline. Half of the energy is stored in the magneticfield, because of incident current, and the other halfis stored in the electric field, because of incidentvoltage. The voltage and current travel in phasewith each other because of the resistive ZC. Onreaching an open circuit the magnetic fieldcollapses, because the current goes to zero. Thechanging magnetic field produces an electric fieldequal, in energy to the original magnetic field. Thenow electric field adds in phase to the existingelectric field, resulting in a corresponding increaseof voltage. at the open circuit to twice the incident-wave voltage. (At this instant a standing wave isdeveloping, because now there is a currentminimum and voltage maximum at the open-circuitterminals, where an instant before, current andvoltage were constant, all along the line.) Theincreased voltage now starts the reflected voltagewave traveling in the opposite direction, as if it hadbeen launched by a separate generator at the open-circuit point.11 Since no energy was absorbed by theopen-circuit load, the returning wave will be of thesame magnitude as the original incident wave. Asthe electric field starts its rearward journey it setsup a new magnetic field in opposite phase to theoriginal, and once more the energy will divideequally between the two fields. The new magneticfield causes the current to build up again to thesame magnitude as before, but in the oppositepolarity, to be relaunched into the line as thereflected current wave (refs. 17. p. 139; 19. p. 4;35, p. 21; 43). The total voltage (or current) at theload at any instant is the sum of the voltages (or

currents) of the incident and reflected waves. Sincethe two currents add to zero at the open-circuitload, the generation of the reversed polarityreflected current wave is verified. The in-phasereflected voltage wave is similarity because thesum of the two voltages at the load is double theincident voltage. The phase angles, θ, of thereflection coefficients at the open-circuit load aretherefore 0 degrees for voltage and 180 degrees forcurrent.

When the load impedance is a short circuitthe reflection-generation process is similar to theopen-circuit case, except that the electric- andmagnetic-field actions and the polarities of thereflected-wave components are reversed. This isexpected when we recall that while current goes tozero in an open circuit, voltage must be zero in ashort circuit. For the voltage to be zero the incidentand reflected voltage waves must cancel oneanother at the load, thus verifying the reversedpolarity of the reflected wave. The correspondingcurrents add to double the incident value, as thevoltages did when the load was an open circuit.The phase angles, θ, of the reflection coefficients atthe short-circuit load are therefore 180 degrees forvoltage and 0 degrees for current. When the load

impedance is a pure capacitance it is equivalent toan additional length of open-circuit line, while apurely inductive load is equivalent to an additionallength of short-circuited line.

When the load impedance containsresistance the reflection will be generated in thesame manner as with an open- or short-circuit load,but it will be less than total, the amount dependingon how much power is absorbed in the resistance.The reflected wave is again generated by thechanging electric and magnetic fields at themismatch point, caused by the change in voltageand current when the incident wave encounters achange in load conditions. Hence the reflectioncoefficient, ρ, is dependent on the differencebetween incident-wave voltage on the line and thevoltage measured across the load.

No reflection arises when the load is a pureresistance equal to the line ZC, because all theincident energy is absorbed and there is no voltageor current variation when going from the line to theload. Thus there is no electric and magnetic fieldchange, no new voltage or current generated, henceno reflected wave.

As the reflected wave propagates back upthe line as a separate and distinct electromagnetic

traveling wave, it encounters only the same low-loss line with resistive ZC

12 encountered by theincident wave in its forward travel to the load.Hence, the magnitudes of both the reflected voltageand current remain constant as the wave plowsrearward, having the same values as when leavingthe reflection generator. (There is a gradualdiminishing effect because of attenuation; this willbe discussed later.) They are, completelyunaffected by the standing waves being developedas the reflected and incident waves, slide past oneanother. The incident voltage and current waves aresimilarly unaffected, continuing in their forwardtravel with constant magnitude until reaching theload.13 Also, as in the incident wave, both thereflected voltage and current pass through zerosimultaneously, and reach their maximum valuesone-quarter cycle later, because the line ZC isresistive. Are not the reflected voltage and currentthen also in phase with each other, like the incidentvoltage and current? Perhaps, but let's not overlookpolarity -- the voltage maximum could be positivewhen the current maximum is negative, in whichcase they would be 180 degrees out of phase witheach other. But does the phase really matter here?Indeed it does - there is probably no otherrelationship more important to the principles ofwave mechanism on a transmission line! Eventhough (the incident and reflected waves travelseparately in opposite directions, they areinescapably related to each other through thecommon line and load characteristics, and theirrespective voltages and currents add vectorially atevery point along the line as the two waves slidepast each other. Hence the polarity, or phaserelationship between the voltage and current inboth sets of waves determines the character of theresultant standing waves, line-input impedance,and any other effect resulting from the vectorcombination. Many aspects of transmission-linephenomena which seem difficult to follow can beresolved rather easily if we understand how thephase, or polarity relationships evolve. So how dowe determine the polarity and how do we establisha reference?

Wave-Travel Analysis

Consider a single wave traveling in a two-conductor line. By following conventional currentflow we can select the appropriate conductor as the

voltage-polarity reference for a given direction ofwave travel which will cause the voltage andcurrent maxima to occur with the same polarity.This polarity relationship may be reversed simply,either by selecting the opposite conductor for thevoltage reference or by reversing the direction ofwave travel. Obtaining the opposite polarity orphase relationship by reversing the conductors is asimple enough concept, but obtaining it byreversing the wave-travel direction has been a pointof confusion for many people.

To reduce the confusion factor, a set ofsimple current-flow diagrams showing both ac anddc treatment is presented in Fig. 3. Use of dc withcenter-zero meters as indicators makes theexplanation of polarity easy. Conventional needlemovement is to the left for negative polarity and tothe right for positive. Once polarity is clear, thebattery may be replaced with the ac generator, andphase will also become clear using the wave formsas indicators. The setups in A and B of Fig. 3 arethe same as in C and D respectively, except that thevoltmeter terminals have been reversed. Observethat the wave or energy-flow direction and voltage-polarity reference selected in A cause line voltageand current flow to be in the same polarity. Nownotice that reversing either the wave direction (asin B) or the voltage reference (as in C) both resultin opposite voltage and current polarities as statedpreviously. Observe that reversing both directionand voltmeter reference polarity (as in D) againresults in line voltage and current flow with thesame relative polarity, though reversed from A. Itmay be helpful at this point to perceive thatchanging the wave or energy-flow direction issimply equivalent to reversing the terminalconnections of the current meter, because thesource changes sides. This is the key tounderstanding the polarity-reversal problem,because for a given voltage-polarity reference thecurrent-flow direction has to reverse when thewave-flow direction reverses.

On the basis of the conditions stated above,if a generator is now placed at each end of a singletwo-conductor line, a reference selected to makethe voltage and current in phase on the line for onegenerator will result in 180-degree out-of-phasevoltage and current for the other generator. This isthe situation which exists with the mismatched rftransmission line -- a source generator at one endand the reflection generator at the other. By

selecting the conventional reference to makeincident voltage and current in phase with eachother (or θ = 0°) it follows that reflected voltageand current must he 180 degrees out of phase witheach other (ref. 35, p. 23).

It is of interest at this point to be concernedwith the nature of the power in the incident andreflected waves. Some writers contend erroneouslythat the voltage-current phase relationship in thereflected wave is 90 degrees. If this was true, thenthe wave would contain only reactive volt-amperes,but no real power. The evidence above disprovesthis contention since we, have seen that thevoltage-current relationship in the reflected wave is180 degrees and not 90. And certainly we willagree that if real power is conveyed in A of Fig. 3,it is also real power in B, or C, even with reversedcurrent-meter or voltmeter terminals. We will agreealso that real power, P equals EI cos θ, in whichcosine θ is the power factor. It matters not whetherthe phase angle is 0 or 180 degrees, for cos 0° = 1and cos 180° = -1. This simply connotes thepolarity difference discussed above. Whenconductor spacing is restricted to the near field, thefundamental principles governing transmission-linepropagation are the same as those which govern allgeneral ac-circuit relations, including electricpower transmission. From these principles weknow that real power flows for every value of θ inall four quadrants, except at 90 and 270 degreeswhere the cosine is zero, yielding zero powerfactor. Wherever the phase is other than 0, 90, 180,or 270 degrees, both real power and reactive, volt-amperes are present. But at 0 or 180 degrees, onlyreal power exists because the absolute value ofpower factor is 1.0 in either case. This clearlyproves that reflected power and incident power areboth real power, and that no fictitious power, orreactive volt-amperes, exists in either one, becausethe current and voltage in the reflected wave arealways mutually 180 degrees out of phase and thevoltage and current in the incident wave are alwaysmutually in phase.

The conflict concerning real vs. reactivepower in reflected waves arises in part fromconfusion between traveling and standing waves,because of insufficient familiarity with both types.To broaden the familiarity, we have logicallyconcentrated first on the traveling incident andreflected waves, because, from the physical viewpoint, standing waves are derived from the

resultant interaction between the two travelingwaves. And thus, sufficient pertinent knowledgeconcerning traveling waves is essential before onecan correctly understand the formation of standingwaves and other correlated phenomena occurringon the transmission line which will becomeapparent as we proceed.

Now that we have a reasonably enlightenedbackground concerning the ingredients; of standingwaves, let us explore the details of theirdevelopment, after which it will be appropriate toreturn to the power conflict for a few briefcomments to clear away any remaining confusion.

Vector Graph Explanation and Standing-WaveDevelopment

The newly launched reflected voltage andcurrent waves, in their rearward travel toward thegenerator, combine with their respective incidentwaves at every point on the line. The continuouslychanging relative phase differences along the linecause alternate cancellation and reinforcement ofthe voltage and current distribution on the line.This results in the formation of the well-knownstanding wave and a change in the input terminalimpedance from the initial line ZC value.

A physical picture of this complexrelationship greatly enhances the understanding ofthe phenomenon. Accordingly, Fig. 4 graphicallyillustrates the progressive phase relations withaccurately scaled vector plots of the incident andreflected wave for visual comparison at every 22.5-degree (sixteenth-wavelength) point on the fine.The reference is from the termination point backtoward the generator. These vector plots, beingsuperimposed circularly around the Smith chart,present certain symmetrical phase-anglerelationships with respect to line length which annot obvious with previous displays.14 (Refs. 2. p.12; 17, p 146: 18. p. 110.) This visual aid, inaddition to providing a new dimension inpresenting the formation of the standing wave,vividly emphasizes the development of capacitiveand inductive components of complex impedance,the understanding of quarter-wave impedanceinverting action, the impedance-repeating phase-inverting action of the half-wave line, and thereciprocal relationship between impedance andadmittance.

For this illustration we have terminated theline with a pure resistance equal to three times theline impedance ZC. Hence a circle representing anSWR of 3.0 is shown. Line length, L, measuredfrom 0° at the termination point (or reflectionplane) toward the generator, is represented byclockwise rotation around the chart.

Current vectors are displayed inside theSWR circle, marked "+" for incident current(magnitude 1.0), “-“ for reflected current(magnitude 0.5), and I for the resultant current.Directly opposite and outside the circle are thecorresponding voltage E vectors. The number indegrees shown with each vector set indicates theangle of the resultant. The vector lengths areproportional to the amplitudes of their respectivevoltage and current waves. By setting the incident-vector length equal to 1.0 the reflected vectorlength, by definition, equals the magnitude of thereflection coefficient, ρ. The reflected-vectorlengths are thus 0.5 because ρ = 0.5 for an SWR of3.0, from the relationship

ρ =−+

SWRSWR

11

(Eq. 2)

While the vector lengths are proportional toeach other, the lengths are not scaled to any chartdimension. These vector plots at each angularposition on the graph contain the necessaryamplitude and phase information to define both thestanding wave and the impedance at the point onthe transmission line represented by the pointwhere the SWR circle intersects the radial line ateach L = 22.5° interval.

The SWR = 3.0 (or ρ = 0.5) circle is basedon the chart scales, and may be observed to have aradius of one half the chart radius. The perimeter ofthe chart has a radius of 1.0, representing totalreflection, i.e., an infinite SWR, The ρ-circle radiusis thus directly proportional to the reflectioncoefficient ρ, so an SWR or ρ circle may beconstructed for any value of reflection by makingthe radius equal to the reflection value, ρ.

In order to make valid phase comparisonsbetween points on the line, wave motion has beenfrozen at an arbitrary point in time, so that allvectors are shown in their true positions, relative toeach other. It makes no difference when the motionis stopped, but the symmetry of the presentation isenhanced if we stop the motion when the incident

vectors at the reflection plane are pointing in thezero direction in the standard polar coordinatesystem,

Observe that at any point on the line theincident voltage and current are always in phase,while in contrast, the reflected voltage and currentare always 180 degrees out of phase, thusillustrating the conclusion of the discussion on thispoint in the previous section on basic reflectionmechanics. At the reflection plane (L = 0°) it maybe seen that all components are in phase except thereflected current, which is 180 degrees from theothers, this is as it should be when the terminalimpedance ties between ZC and an open circuit.

As we travel clockwise from the reflectionplane toward the generator, each incident and eachreflected vector rotates the same number of degreesas the change in position along the line. Butobserve carefully, for this is very important: Theincident-wave (+) vectors rotate counterclockwise(phase leading), while the reflected-wave (-)vectors rotate clockwise (phase lagging). Forexample, at 45 degrees from the termination theincident voltage vector is at +45 degrees, while thereflected voltage vector is at -45 degrees, for a totalphase difference of 90 degrees. Thus, for everydegree of motion along the line the relative phaseangle between the incident and reflected voltagechanges two degrees.15 This can be readilyunderstood when we consider that in the distancefrom the reflection plane to our observation pointthe reflected wave has traveled twice as far as theincident. From the observation point the incidentwave travels only in the reflection plane, while theportion of the incident that is reflected travels anequal additional distance in returning to theobservation point.

Now let's see what happens at L = 90°, orλ/4 of travel from the load. At the load, where L =0°, the incident and reflected voltages are exactlyin phase with each other, giving the reinforcementmentioned earlier (resultant = 1.5). But at 90degrees toward the generator the two voltagevectors have each rotated 90 degrees in oppositedirections and are now 180 degrees out of phaseand opposing (resultant = 0.5). Going on to L=180°, or λ/2 from the load, we see that each voltagevector has now rotated 180 degrees, but in oppositedirections of rotation. Hence the vectors haverotated 360 degrees relatively, and once again areexactly in phase with each other and reinforcing.16

At the points between L = 0° and L = 90°,the resultant voltage vector, E, is seen to diminishgradually from the maximum of 1.5 to a minimumof 0.5 and then increase back to the 1.5 maximumat L = 180°. In Fig. 5 these resultant magnitudevalues at each point have been plotted on the morefamiliar rectangular coordinate graph. The smoothcurves connecting the plotted values indeed yieldthe familiar standing-wave pattern. The curvesverify the relationship

SWR =+−

11

ρρ

= 3.0 (Eq. 3)

by first adding to and then subtracting ρ - 0.5 fromthe incident voltage 1.0.

Line lengths greater than a half wave areaccommodated merely by continuing on around thecircle again (Fig. 4), repeating the some valuesencountered 180 degrees earlier, thus establishingthe periodicity of the standing wave. Only losslessline is being considered here; correction factors forattenuation, which change the circle into a spiral,will be presented later. The basis for the phase- orpolarity-reversing characteristic of the 180-degreeor λ/2, line may be observed on the vector graph bynoting that the specific phase of the voltage vectorsat the 180-degree point on the line is 180 degreesfrom their phase orientation at the reflection plane(L = 0°).

The significance of the constant 180-degreephase difference between the reflected voltage andcurrent will emerge if we now compare the phaseand magnitude of the current vectors with thevoltage vectors previously studied. We see that atthe reflection plane (L = 0°), where the incident andreflected voltage waves are in phase and adding tocreate a voltage maximum, the correspondingcurrents are out of phase and opposing to create acurrent minimum. And at L = 90°, where thevoltage waves are out of phase and opposing for aminimum, the currents are in phase for amaximum. Which graphically illustrates why themaxima and minima of the voltage standing waveare ALWAYS separated by 90 degrees from thecorresponding maxima and minima of the currentstanding wave. This phenomenon is caused directlyby the 180-degree phase difference between the

reflected voltage and current, as strikingly shownby the vector display.

A visual comparison of the angularpositions and magnitudes of both the voltage- andcurrent-vector resultants on the vector graph (Fig.4) may be made with the corresponding positionsalong the plot of Fig. 5. This comparison willenhance the understanding of this importantconcept. It is important because as we proceed itwill be seen to be the basis for the impedance-transforming properties of the transmission line,including the quarter- and half-wave sections,which are only two specific conditions of thegeneral case.

As we discuss the concept of impedance inthe next section and solve matching problems lateron, conjuring up a mental image of the actionoccurring on the line will at times be more helpfulin understanding the difficult points than throughlogical reasoning alone. It is especially important tohave a clear image of the formation of the reflectedwave, because the reflected wave must beconsidered as a separate traveling wave identical tothe incident wave, except for the direction and,usually, the magnitude. This point is important,because it helps us keep in mind that the reflectedvoltage and current waves travel the line 180degrees out of phase with each other, and thustransfer actual real power during their travel. It isessential, to the process of reinforcement andcancellation of voltage and current along the line inthe formation of the standing wave that real powerbe conveyed in the reflected wave, as if it had beendeveloped by many tiny little generators at everypoint all along the line. It will also become clear inthe section to follow why the impedance along aline changes in the presence of reflections only

because real power is flowing, in both directions.(refs. 2, p. 70;35, p. 24; 42).

We are stressing this point because, asmentioned previously, some writers have presentedthe erroneous viewpoint that the reflected waveconveys no real power, with the argument that thereflected voltage and current are 90 degrees out ofphase with each other and are therefore wattless.The reflected wave would indeed convey zeropower if its voltage and current were 90 degreesout of phase, but the, argument is incorrect becausethey actually travel 180 degrees out of phase, as wehave previously shown. At least two writers,W9IK17 and W5GO,18 would have us believeotherwise. It is easy to reach the wrong conclusion,however, because of the lack of a clear image ofthe reflection process and the somewhatcomplicated wave mechanics on the line. It may bethat W9IK has confused the reflected wave withthe standing wave, which both writers may haveconfused reflected voltage E and current I with linevoltage E and current I.19 The true nature of thereflected wave as a separate electromagnetictraveling wave must be appreciated. It is interestingto note that W9IK states, ". . . re-reflection ofpower at the input end (of the line) is impossible toaccept since the necessary conditions of impedancemismatch are not present," and yet his Fig. 2 showsa, means for obtaining a (conjugate) match. This isalso true in his Fig, 1, since he implies that his finalamplifier is loaded and tuned properly. Now thevery essence of the conjugate match is its totallyreflecting mismatch for waves traveling toward thegenerator, while presenting a perfect match forwaves traveling toward the load. Yetmisunderstanding of this important basic concept iswidespread among the amateurs. This concept isbasic to the operation of passive frequency-selective filters. and the mechanism for developingthe conjugate match simply constitutes a filter ofthis type. The mechanism behind its operationinvolves wave interference and reflections, whichwill be described in detail shortly, again using thevector graph as a visual aid.

It is understandable that the 90-degreedifference of position on the line which existsbetween the voltage and current maxima (orminima) of the standing waves (described a fewparagraphs earlier and shown in Fig. 5) could havebeen mistaken as the phase difference between thereflected voltage and current. And the similarity

between line voltage and current behavior with thevoltage-current relationship in ordinary ac circuitryalso makes it easy to understand why line voltage Eand current I are being confused with the reflectedvoltage E and current I. This is because, in additionto the in-phase line voltage and currentcomponents, which convey only the net powerflow, line voltage and current do contain reactivecomponents which are 90 degrees out of phase witheach other when reflections are present. Obviously,these reactive components convey no real power.Unfortunately, some people who are well versed inac circuitry using lumped constants, but who areless familiar with transmission-line operation,sometimes make the error of assuming that the twocircuit types are identical in electrical performance,and so we should be wary of making unwarrantedcomparisons which can bring about disastrousconsequences of the type we are attempting tostraighten out here.

Another specious argument set forth is thatreflected power cannot be real power because itcannot perform work. We will prove this argumenterroneous by showing how power in the reflectedwave can do work. Now a simple rf voltmeter andammeter in the line will indicate only the resultantvoltage and current of the combined Incident andreflected waves (E and I on the vector graph), theproduct of which, when weighted by the cosine ofthe phase angle between them, yields only theresultant, or net power flowing toward the load.But there are many devices in common every-dayuse which selectively extract either the reflected orthe incident wave (or both) from the line separately(independently of the standing wave), and thuspermits separate measurement and analysis of thepower associated with the waves traveling in eitherdirection. One such device is the directionalcoupler. Another is the circulator. This is a three-port directional device in which the reflected wavefrom a mismatched load on the second port iscompletely diverted away from the input feedlineand emerges from the third port. The reflectedwave cannot get back onto the feed line to interactwith the incident wave to develop a standing wave,and thus does not change the line-input impedanceat the source feeding port one. However, currentflow through a resistor placed on the reflected-power-output port (port three) develops heat (I2R)equal to the amount of the power reflected from themismatch at port two. A four-port hybrid coupler

can be connected to perform in the same manner asthe circulator. Directional rf devices most familiarto amateurs are the simple reflectometer SWRindicator and the directional wattmeter (ref. 18, p.180; 38; 40; 42). The meter in either one isactuated by rf power -- absorbed from one of thetraveling waves on the line -- either the forward orreflected, as selected. If the reflected wave werewattless reactive power, no power would beavailable to actuate the meter movement in theSWR indicator, or to produce heat from the currentflow in the resistor on the third port of thecirculator. Furthermore, for power to becomewattless on being reflected would violate the mostgeneral and fundamental of all physical laws,namely the law of conservation of energy (ref. 35.p. 25), On the basis of this law, if all of the energyflowing in the line toward the load cannot beabsorbed in or dissipated by the load, that portionwhich is not absorbed must appear somewhere. Itcannot just disappear or cease to exist as if bymagic. The reflected power recovered as heat in thecirculator is a typical proof of this fact.

Here is another way of expressing netpower flow through the line, which enables us tobreak power down into its incident and reflectedpower components. The expression is obtainedfrom the power formulas, e.g.,

P EI= (Eq. 4)

or PE E

Zc

=×max min (Eq. 5)

On rf lines, power P E= = ×max min I (Eq. 6)

Or

PZc

=E x Emax min

Now Emax = (produced by E++ E¯) occurs as shownin Fig. 4, where the incident and reflected voltagesare in phase at L = 0 and 180°, and Emin (producedby E+ - E¯) occurs where they are 180 degrees outof phase at L = 90°, Later we shall see that both theresultant voltage E and current I are nonreactive atthese points on the line, while being reactive

everywhere else along the line between the points.Because these points are nonreactive, the productsof their voltages divided by the line impedance, ZC,yields the net power flow exactly. But recallingthat E+ and E¯ are always nonreactive, we canreplace the term Emax by E+ + E¯ and Emin by E+ - E¯,and thus:

PZc

=E x Emax min

( ) ( )

=+ × −+ − + −E E E E

Zc

(Eq. 7)

Multiplying out the numerator terms gives thedesired incident and reflected components:

PE

Z

E

Zc c

= −+ −2 2

= net power flow (Eq. 9)

The first term on the right of the P expresses thepower associated with the incident wave and thesecond term the reflected power. This simpleseparation of power into two separate components,each associated with one of the traveling waves,can be done on a lossless or low-loss line, wherethe ZC is resistive. If the line has appreciable lossthe interaction of the two waves gives rise to a thirdcomponent of power which we can disregard, aslines normally used by amateurs are usually in thelow-loss category. (See refs. 18, p. 150, and 37, p.129.)

This separability of the forward andreflected powers forms the physical basis for theoperation of reflectometers and directionalwattmeters, (ref. 38) in which either the forward orreflected component is sensed by taking advantageof the 180-degree out-of-phase relationship of thereflected components of voltage and current whilethe forward voltage and current components are inphase with each other. In these operations a sampleof the voltage across the line is added to a sampleof a voltage derived from the current in the line.When the amplitudes of the samples are adjusted tothe correct relationship (determined by lineimpedance ZC), the two reflected componentscancel, so that the sum represents the forwardcomponent alone. By reversing the phase of thecurrent sample 180 degrees, the forward

components cancel and the resulting sumrepresents the reflected components alone. A meterconnected to indicate the voltage sums can now becalibrated in power, because the square of thevoltage is proportional to power. Thus, thebeautiful aspect of the directional wattmeter is that.when properly calibrated, it indicates the truepower in the transmission line with the lineterminated in any load impedance; the load can beeither a match or a mismatch, and it can be reactiveor nonreactive. The meter does this because theforward power value is always equal to the sum ofthe line-input power plus the reflected power; thusit indicates the true power which is actuallyincident on the load. In the reflected-powerposition the meter indicates the amount of theincident power which was not absorbed by theload, but which adds to the line-input power fromthe transmitter at the line input, or at whateverpoint in the line the conjugate match is performed.(Ref. 18, p. 191.) The difference between theforward and reflected readings, is, thus, the netpower flow in the line at whatever point thewattmeter is inserted. In a lossless line the net

power flow indicates the line-input power, which isthe absorbed power exactly; the two are identicalanywhere on the line. In a line with attenuation themeter indicates the line-Input power if it is placedat the line input, or it reads the absorbed power if itis placed immediately ahead of the load. Thedifference between these readings is related to theline attenuation. Of course there may be, forpractical reasons, errors in the actual results fromSWR measurements -- diode nonlinearity atvarious power levels, for one example (ref. 40).

Except for a somewhat different viewpointconcerning the nature of reflected power, the workof W6EL, formerly K6CYG. closely parallels thebasic theme of this series of articles and reaches thesame conclusions (ref. 44). Reference to Shallon'swork was omitted from Part 1 of this series becauseof author error, but his work deserves thoroughreview at this point.

A detailed explanation of why the pointsmentioned above take place and of the wavemechanics of conjugate matching will appear inPart IV of this series, in a subsequent issue of QST.

Part 4 - A View into the Conjugate Mirror

IN PART 3 some basic concepts werepresented concerning reflection generation, wavepropagation along the line, and the development ofstanding waves. Then it was shown mathematicallyhow net power is separated into its incident andreflected components (on lossless and low-losslines), after which it followed logically to explainhow the power separation is physically realized bydirectional devices, such as a directional wattmeter.While learning about wattmeter operation and howto interpret the indications, we saw that the incidentor forward power in the line between, the matchingpoint and the load is greater than the powersupplied by the source generator when the line isterminated in a mismatched load. In this part wewill explore this situation in detail, because it is ofconsiderable importance to the amateur since itrelates directly to the operational flexibility of hisantenna system. Appreciation of the fundamentalsinvolved in this seemingly anomalous situation willfree him from the prevalent notion that he isrestricted to operating with little or no mismatch atthe antenna/transmission-line terminals.

The explanation of directional wattmeteroperation in Part 3 should help in understandingwhy the incident power appearing on the linebetween the matching point (such as a linematching network) and the mismatched load can behigher than what the transmitter can supply. This isa normal condition which must exist in order for amismatched load to absorb all the power deliveredby the source,20 while at the same time reflecting apercentage of the total power it receives. To dothis, the load must receive more incident powerthan what is supplied by the transmitter. The basisfor understanding this rather subtle concept lies inthe wave mechanics behind the principles ofconjugate matching introduced in Part 1 anddefined in Part 2. The wave aspect of this subjecthas been presented in the literature (known to theauthor) only by Slater (ref. 35) and Alford (ref. 39).Perhaps this restricted exposure may account forsome of the confusion in this area among engineersand amateurs alike. For example, the many "cookbook" recipes and graphical directions for stub-

matching a mismatched line tell "how" to do it, butoffer little insight toward visualizing the wavemechanics through which the match isaccomplished. This insight, however, goes to theheart of the transmitter-to-line coupling problem. Itclarifies how the reflected wave becomesrereflected at the matching point. If the matchingdid not produce this effect, the reflected wavewould travel back to the generator, and would thusreduce the amount of the power made available bythe source generator (ref. 19, p. 37).

Reflection Mechanics of Stub Matching

An introduction to the reflection mechanicsinvolved in conjugate matching concerns conceptsof line-input impedance and angle of reflectioncoefficient, which will be explored in detail in alater section. (The reflection coefficient angle wasintroduced briefly in Part 3, para. 1, and footnote15.) As stated with the definition of the conjugatematch in Part 2, the matching is accomplished byinserting a nondissipative mismatch at the matchpoint; this produces a complementary reflectionwith which to compensate, and cancel, the wavereflected from the mismatched load. It was alsostated that conjugate matching conditions can besatisfied by a correct adjustment of either the finaltank tuning circuit (ref. 4, Part 3), or of a line-matching network if one is used. Because stubmatching uses the identical principles and is easierto visualize, we will use its technique todemonstrate the wave mechanics. In stub matching,the stub provides what seems like an anomaly -- anondissipative discontinuity, or mismatch. Whilewe usually think of the stub as providing a match,rather than a mismatch, we will discover that theconjugate match results from the mutualcancellation of two complementary reflected wavesgenerated by two complementary mismatches. Onewave is that reflected from the terminating loadmismatch, and the other is a new reflected wavegenerated by the stub mismatch, equal to the load-reflected wave in both magnitude and phase, but of

opposite phase sign. Wave interference betweenthese two complementary waves at the stub pointcauses a cancellation of energy flow, or a null inthe generator direction resulting from the differencebetween the two reflected waves, and an energymaximum in the load direction comprising the sumof the two reflected waves and the source wave.The effect of the wave-interference cancellationpresents a virtual one-way open-circuit21 to wavestraveling toward the generator, which blocks boththe load-reflected wave and the stub-reflected waveat the stub point from any further rearward travel.These waves are totally reflected toward the load,being in phase with the incident wave. Thus, thereis complete cancellation of the effect ofdiscontinuities (such as the stub) and the reflectionsof waves traveling toward the load.

This wave-interference mechanism whichaccomplishes the matching will become moreevident as, we investigate the reflection coefficientsof the two mismatches with the aid of examplesusing the Vector Graph (Fig. 4, Part 3). There weshow a load of 3 + j0, which gives a three-to-onestanding wave ratio along the whole line.22 TheVSWR of 3 is shown by the dark concentric circle,which is the locus of impedances anywhere on theline for a load of 3 + j0. This circle intersects theresistance circle marked 1.0 at the two points, Aand D. Therefore at these points along the line theresistive component of the impedance equals 1.0times ZC, which is the desired point for attaching amatching stub. Points A and D are also intersectedby a reactance circle. At point A the reactance isnegative, 1.15 times ZC, and at point D thereactance is positive, also 1.15 times ZC. Theconditions for obtaining reflected-wavecancellation by wave interference are the well-known stub-matching requirements as follows: (1)a stub is placed where the line resistancecomponent equals the line characteristic impedanceZC (such as at points A and D), and (2) the stubreactance is made equal in magnitude and oppositein sign to the line reactance (resulting from thephase relationship between the incident andreflected waves at the stub point) so that thereactances cancel to zero, (refs. 2, p. 116; 19, p.97). This sounds almost like the conjugate-matchdefinition itself, doesn't it? The correct point forinserting a stub in series with the line nearest theload is at point A or D on the VSWR circle; anyhalf-wave interval from these points further from

the load may also be used since the impedances arerepeated every half wave on the diagram (and onthe line). These examples show series stubs topermit impedance treatment for clarity. (Whileparallel or shunt stubs are used more prevalently,analysis using the shunt form would requireadmittance treatment.)

We will now see how reflections add at amatching point to produce the matching effect.We'll also see why a directional wattmeter will givea true reading of incident power between thematching point and the load which is greater thanthe power supplied by the transmitter, when theline is terminated with a mismatched load. A littlelater we'll also see how these principles apply topractical feed-line matching networks.

At point A, which is 30 degrees from theload (at L = 30°), the unmatched voltage reflectioncoefficient is ρ E = 0.5 ∠ - 60°. This means that thephase of the reflected voltage wave lags theincident wave by 60 degrees at point A. The lineimpedance E/I at this point is 1 - j1.15. A matchcan be effected by connecting an inductivereactance, such as a stub or a lumped inductance of0 + j1.15 in series with the line at point A. Now,the reactance-cancellation effect of the positivereactance stub on the equally negative reactance ofthe line is generally understood, but several pointsare not always clear regarding the effect on thecomponent waves: What characteristics of the stubcause it to counteract the reflections from the load;also, why does the stub cause the incident power torise between the matching point and load'?

In answer to these questions, let's determinefirst the reflection coefficient produced by the stubif it were inserted in a perfectly matched line. Inthis condition we may analyze the stub generatedreflection in the absence of any other disturbanceor reflection on the line. If we station ourselves juston the load side of the stub point with the stubattached and look, into the line toward the matchedtermination, we will see a pure resistance equal tothe line characteristic impedance, ZC. We knowfrom matched-line theory that if we remove the lineportion extending from the stub to the load andinsert the terminating resistance R = ZC directly inseries with the stub across the open-ended line, wemay look into the line toward the stub from thegenerator and see the same conditions of reflectionas were present before the line section wasremoved. Thus the series circuit comprising the

matching resistor and the stub performs as a 1 +j1.15-ohm mismatched load terminating the line.Precisely this same reflection will be produced nomatter where the stub is inserted in a matched line.The Vector Graph shows this impedance of 1 +j1.15 to appear at point D, for which the voltagereflection coefficient is ρ E = 0.5 ∠ + 60°. Note thatthis is the same magnitude and phase but ofopposite phase sign to the reflection appearing atpoint A resulting from the load mismatch (3 + j0).

Thus the stub mismatch produces the samemagnitude of reflection (and the same SWR) aswas produced by the load mismatch, but the stub-reflected voltage wave leads the incident wave by60 degrees, while the load-mismatch wave lags by60 degrees. If the stub is now attached at thematching point (corresponding to L = 30° on theVector Graph) with the 3 + j0 load terminating theline, both the stub- and load-mismatch reflectionswill be produced simultaneously. As a result of

their opposite-sign phase relationship, the leadingstub-reflected wave and the lagging load-reflectedwave cancel each other at the match point. The,voltage reflection coefficients of load and stub thusadd vectorially to zero degrees (ρ E = 0.5 ∠ 0°),which tells us that the resultant of the tworeflections is exactly in phase with the incidentvoltage wave at the match point as shown in Fig.6A. (The amplitude resulting from thistrigonometric addition will be considered later, butknowledge of these angular relationships shouldadd to the appreciation of the mechanics of line-reactance cancellation by the stub.)

Now that we know what is happening withthe voltage waves we also want to investigate thecurrent waves to learn about the impedancerelationship at the matching point. As definedearlier, reflected current is 180 degrees out of phasewith reflected voltage, so the current coefficient isfound on the Vector Graph 180 degrees away ordiametrically opposite the corresponding voltage-coefficient point. Thus we find the currentreflection coefficient for the load mismatch at pointC with ρ I = 0.5 ∠ +120° Similarly, the stub-

mismatch current coefficient found at point B is ρ I

= 0.5 ∠ -120°. Note that the current coefficients ofload and stub are also of equal magnitude andphase, but of opposite phase sign. But while thevoltage angles add to 0 degrees, the currentcoefficients add vectorially to ρI = 0.5∠ 180°. Theresultant current reflected wave is then 180 degreesout of phase with the incident current as shown inFig. 6B. So we have incident and reflected voltagesin phase, and incident and reflected currents out ofphase -- and the wave arriving at the match pointfrom the generator sees a perfect match.

These facts portray a significant message.In the opening paragraphs of Part 3 the wavemechanics involved in a line terminated by an opencircuit were described. There we learned that thereflection coefficient angle of the voltage is 0degrees and of the current is 180 degrees. With anopen-circuit condition, the unabsorbed voltagewave which is incident on the termination isreflected with no change in phase, while theunabsorbed current wave is reflected with a phasechange of 180 degrees, a complete reversal ofpolarity. The reflected-wave phase relationships atthe match point which we established above (by thevoltage and current resultants of the stub- and load-

mismatch waves) indicate precisely the sameconditions that prevail in a line terminated in anopen circuit, so far as the reflected waves areconcerned (but not the incident wave.) Therefore,the effect of the two reflected waves arriving at thematch point is to establish an open circuit to thewaves generated by the two mismatches. Thus,these waves become totally reflected at the matchpoint and undergo open-circuit phase-changerelationships as described above. The resultantreflected voltage wave thus does not change phaseduring this reflection; remember that it was alreadyin phase with the incident wave prior to itsreflection. The resultant current wave changesphase by 180 degrees on reflection, and because itwas 180 degrees out of phase with the incidentcurrent wave just prior to its reflection, the present180-degree reversal now places it also in phasewith the incident current wave. Now that bothvoltage and current rereflected waves are in phasewith their corresponding incident waves, additionof the voltages and currents occurs at the reflectionpoint. Thus the conclusion: The power contained inthe reflected waves adds to the incident power.

Before we proceed any further, let usconsider that the above conclusion was based onthe assumption that an open-circuit condition existsat the match point for the reflected waves travelingtoward the generator. The assumption was based onthe similarity between the reflection coefficientswhich we established at the match point by waveinterference and those known to exist at an open-circuit termination. We can verify this assumptionby an alternate method, based, for example, on theopen-circuit magnetic-field theory from Part 3.

Let us first observe the net value of allcurrents flowing at the match point at the instantthe two reflected currents form their resultant θ =180°; at this instant we will see an initial suddendrop in resultant line current I because of wavecancellation as the reflected-wave resultantbecomes aligned exactly out of phase with theincident current. This drop is shown graphically inFig. 6B, where the original resultant current I (aswith no matching stub present) suddenly drops tothe new instantaneous resultant value I' from theeffect of the stub discontinuity. Now recallingbriefly from the open-circuit field theory presentedin Part 3, para. 2, we know that when current drops,the magnetic field also drops. The changingmagnetic field produces an electric field equal to

the energy reduction in the magnetic field. The newelectric field adds in phase to the existing electricfield, producing an increase of voltage at the matchpoint. This increase in voltage now starts a wavetraveling in the opposite direction, which isactually now in the same direction as the incidentwave, thus adding to it. The increased electric field(now an enlarged incident electric field), as itmoves toward the load, produces a new magneticfield equal in magnitude but of polarity opposite tothat of the original field. This new magnetic fieldnow causes current to build up again to the samemagnitude as the original reflected current, but ofopposite polarity and direction. Thus the newcurrent wave is now also traveling in the samedirection with the same polarity as the incidentcurrent wave, adding to it and enlarging it just asthe rereflected voltage wave added to and enlargedthe incident voltage wave.

By following these field-current-voltagereactions through their natural sequence of events,it can be seen that we have obtained the sameconclusions as those previously obtained, thusjustifying the assumption that the resultantreflection coefficients at the match point havedefined an open circuit to the reflected waves. Theexistence of the reflectance at the matching point istherefore verified, with the result that both thereflected voltage and current have indeed beenrereflected and the power associated with them hasthus been effectively added to the power containedin the incident wave. Thus when the line isterminated in a mismatch, causing reflected powerto exist on the line, the sum of the source andrereflected powers (which is traveling only towardthe load) must be greater than the power deliveredby the generator alone. And since we have shownhow the stub acts to counteract reflections from theload, our original questions concerning the stubcharacteristics have been answered.

With a 3:1 SWR, where ρ = 0.5, 100 wattssupplied by the transmitter will yield 133.3 watts ofincident and 33.3 watts of reflected power.Neglecting losses, 100 watts will also be absorbedin the load. From Fig. 4, reflected power ρ2 = 0.25,or 25 percent of the incident power, leaving 75percent absorbed by the load (1 - ρ2 = 0.75).Incident power is 1 /(1 - ρ2) times the suppliedpower, so 1/0.75 = 1.333, and 1.333 times 100watts equals 133.3 watts.

In a typical realistic case where the flat-lineattenuation is 0.50 dB (corresponding to 175 feet ofRG-8/U at 4 MHz, 85 feet at 14 MHz, or 85 feet ofRG-59/U at 4 MHz), if the load were perfectlymatched to the line (1.0 SWR) the 100 wattsdelivered would be attenuated to 89.13 wattsduring travel to the load. But with a 3:1mismatched load the additional one-way lineattenuation (because of the SWR) is 0.288 dB. Theincident power at the conjugate-match point wouldthen be 124.78 watts (0.288 dB below 133.33watts), and 111.21 watts of power (0.5 dB below124.78 watts) reach the load; 27.80 watts (25percent) are reflected, leaving 83.41 watts to beabsorbed. Of the 27.80 watts reflected, 24.78 wattsarrive back at the input to join the 100 watts ofsource power to develop the 124.78 watts ofincident power. The 5.72 watts difference betweenthe power absorbed in the matched and the 3:1mismatched load (0.288 dB) is insignificant.Information on calculating these values will bepresented later. These values are typical of dataobtained during actual routine measurements in aprofessional laboratory. They provide additionalevidence that reflected power is real and notfictitious; if it were fictitious power, no more than66.85 watts (75 percent of 89.13 watts) would beavailable to the 3:1 mismatched load. But the 83.41watts actually absorbed is 93.58 percent of theamount absorbed in the matched load, the loss of6.42 percent being completely accounted for in lineattenuation alone.

Matching Networks and Reflection Mechanics

We now wish to delve further into thewave-interference principles demonstrated usingthe stub technique, to learn how the principles alsoapply to both resonant quarter-wave seriesmatching-transformer operation and the typicalamateur "antenna tuner" (line-matching network)or Transmatch. In order to visualize the inherentgenerality of these principles we need to developsome additional concepts concerning stub matchingand embark on a somewhat different line ofreasoning. As may be surmised from the examplepresented above, the fundamental principle behindthe elimination of reflections is to have eachreflected wave canceled at the point where theelimination of the reflection is desired byinterference from another wave of equal magnitude

and phase but opposite phase sign (ref. 35, p. 58).A transmission line of the appropriate length whichhas one end effectively open circuited and the otherend short circuited possesses the reflection-producing characteristics required to developcanceling waves of the correct phase in relation tothe wave to be canceled.

Canceling waves can be developed by usingother line arrangements, but for the purpose ofdemonstrating the principle, we will use thearrangement just stated, which, as shown in Fig. 7,shows how a stub performs the matching functionin practice. Fig. 7A is the conventionalrepresentation of a typical series-stub circuit (usingthe values of our previous SWR = 3 example), inwhich section F is called the feed line, section S isthe stub, and section T is an impedance-transforming section which we will call thetransformer. We'll now discuss these in greaterdetail. To clarify the approach, Fig. 7A is redrawnin Fig. 7B, with the stub and transformer shown asone continuous straight-line section. This straight-line section will presently come to life as the heartof the wave-interference-producing mechanismfound in all stub-matching operations. This isbecause its physical length will be adjustedarbitrarily so that waves reflected at each end willreturn to the feed point with equal magnitude andphase, but with opposite phase sign.

We have seen earlier that a voltage wave isreflected with zero phase change from an opencircuit (or from any resistive termination greaterthan a matched load), and is reflected with 180degrees of phase change from a short circuit (orfrom any resistive termination less than a matchedload). With a current wave the inverse is true. Sofrom the viewpoint of reflection behavior, one endof the straight-line section will be considered asbeing open circuited and the other end shortcircuited; which end will be open and which endshort circuited will depend on the character of theload. In our example in Fig. 7B the load endbehaves like an open circuit as far as wavereflection is concerned, while the other end (thestub) is short circuited. The action occurring in thisline section in the process of developing theinterfering wave-canceling relationship is asfollows. Either a voltage or current wave isassumed to enter the line section at the feed-lineentry point. The energy divides, one portion of the

wave traveling toward one end of the section, andthe other wave portion traveling toward theopposite end. After each wave portion encountersone reflection the returning waves will each havethe same absolute value of phase but opposite sign,or polarity, on return to the point of entry.

The opposite phase polarity between thetwo reflected waves (arriving from oppositedirections) results because reflection at one end isaccompanied by a 180-degree phase reversal, whilereflection from the other end is not. As statedabove, the phase reversal of one wave but not theother is caused by the opposite conditions ofreflection at the two ends of the line, one end open-and the other end short-circuited. Note in Fig. 7that in each case, the phase of the reflected waves(of both voltage and current) is of opposite polarityon opposite sides of the feed line as they returnfrom the stub and load directions. The wave entrypoint, where the feed line is attached, is thematching point, and divides the line section into itstwo complementary portions: the stub portion, S,and the impedance-transformer portion, T.Electrically, each portion is the complement of theother, because the waves reflected from the end ofeach portion returning to the match point arecomplementary in phase relationship and equal inmagnitude. Herein lies the basis for the termcomplementary mismatches as used earlier, becauseeach portion presents a complementary mismatchto the feed line.

We will see a little later that thiscomplementary mismatch concept is of greatimportance to matching in general, because thecomplementary relationship holds no matter wherethe feed-line entry point is positioned on the stub-transformer line section. The importance prevailsbecause the canceling wave and the reflected waveto be canceled will be of the same magnitude andphase but opposite phase sign at whatever point onthe quarter-wave line section the feed line isattached. This is true with two provisions: (1) thecharacteristic impedance ZC of the feed line F mustbe the same as the resistive component of thetransformed impedance appearing on thetransformer line at the feed point, and (2) the lengthof the stub portion S must be adjusted to produce areactance equal and opposite in polarity to the linereactance appearing at the feed point. The lengthmay be found from the expression

SjXZc

L = arctan (Eq. 10)

where SL is the stub length in electrical degreesjX is the line reactance (obtained from the

Vector Graph)ZC is the characteristic impedance of the

stub section. (ZC = 1.0 here, becausewe are using normalized impedances.See footnote 22.)

The transformer section T transforms the load tovarying values along the line. Hence, for a propermatch, the magnitude of the F feed line impedance,ZC, depends on the location of the feed point andvice versa. This concept is not generallyappreciated, and it is certainly not readily apparentfrom the usual stub-length and position-indicatinggraphs appearing in many publications.

Stub Matching Versus Network Matching

We are now getting closer to seeing howstub-matching principles extend to line-matchingnetwork operation. If we look further into thereflection characteristics of what have generallybeen considered to be different techniques ofmatching, a fascinating revelation of the similaritybetween all of these various techniques willemerge; stub, hairpin, λ/4-series transformer,Transmatch, L network, and so on, are all in thiscategory. And there is a logical reason for thissimilarity; these techniques all have one essentialingredient in common -- reflections! Reflection andmatching are applied in transformers that rely onreflections from the end terminals, where a changein impedance level exists. As we discussed in Part3, any abrupt change in impedance level appears asa discontinuity to the smooth flow of theelectromagnetic wave, and results in producing areflection. The transformer accomplishes the taskof matching its input and output impedances bycontrolling the phase and magnitude of the wavesproduced by reflection at its boundaries, or endterminals, so that all the reflections produced ateither end are canceled by those arriving from theother end (ref. 35, p. 58). This is what was meant inthe reference to "controlled reflections" in Part 1,para. 3. A corollary to the seeming anomaly of thestub producing a mismatch instead of a match, is

that we match to avoid reflections, but we can'tmatch without them when different impedancelevels are involved.

From Eq. 1, Part 3, we know that the load-reflection magnitude is determined by the ratiobetween the load impedance ZL and the lineimpedance ZC of the transformer. So in furtheringthe understanding of the role played by reflectionsin the process of impedance matching, it isinteresting to make two additional observations onthe Vector Graph. First, the reflection magnitude,or SWR, determines the position on the transformersection T where the resistance component of lineimpedance E/I equals the line ZC. This position, werecall, is the matching point, and fixes the length ofthe transformer section T. In making thisobservation remember that the diameter of theSWR circle is proportional to the VSWR. By

tracing along the R = 1.0 circle we can see that asthe diameter of the SWR circle changes, the pointwhere the R = 1.0 circle and the SWR circleintersect moves accordingly. A radial line drawnthrough this intersection point, and extending to theline-length scale L, will thus indicate the angulardistance (T) from the load to the matching point for

a given SWR. We recognize this observation assimply the conventional method of using the SmithChart for determining the stub position when thetransformer and feed-line impedance ZC are equal.But second, using a radial indicating line in asimilar fashion while tracing along the SWR = 3.0circle as it intersects the various other resistancecircles, we see that for a given SWR the resistancecomponent of the line impedance E/I changes withposition along the transformer. These twoobservations together reveal a flexibility availablein the approach to a matching design that providesa step toward visualizing the fundamentalsimilarity of the different matching techniques.This flexibility includes the following threeconditions, which will be explained in more detail:

1) There is no restriction on thecharacteristic impedance Z C of the transformersection T that requires it to be of the same value asthe F feed-line impedance Z C -- it can range fromlow (coax) to high (open-wire line).

2) The length of the transformer sectionhaving a given ZC can be found which willtransform the resistance component of theimpedance to the value of a matching F feed-lineimpedance ZC which differs from the transformerZC. However, the transformer section can have alength which is not limited to the distance from theload to the first point at which the resistancecomponent equals the feed-line ZC. The transformercan extend from the load to either of the two pointswhere the resistance component is seen to equal thefeed-line ZC on the SWR circle, or any electricallength extending beyond these points by an integralmultiple of a half wavelength. (We will see laterhow the use of the Transmatch or an L networkassists in obtaining the required electrical length,and thus removes all restrictions from any specifiedphysical length of transformer, ie., from the loadall the way to the operating position.)

3) The action of the stub portion can beperformed by any nondissipative reactance of theproper value, whether by a lumped-constantcomponent, or by a separate line section of anyreasonable ZC value which has the proper length topresent the required value of reactance. Theelectrical length of the stub is always directlyrelated to its reactance. Now we will see in terms ofwave or reflection mechanics how matchingobtained by the various techniques recitedpreviously is described by these three parameters:

transformer impedance, transformer length, andstub reactive elements.

In our earlier example using the stubtechnique the magnitude of the reflectionsappearing at each end of the transformer sectionwas the same (0.5, or an SWR of 3:1 for the load atone end and the stub at the other). In other words,the magnitudes of each complementary mismatchwere identical. For the present, we will retain thecharacteristic impedance ZC = 1.0 for the entirestub-transformer line section, but based onconditions 1 and 2 above, we may change the feed-line impedance as conditions dictate. Consider nowthe effect of increasing the length of thetransformer section and shortening the stub sectionin accordance with equation 10. For example,while referring to Fig. 7C and the Vector Graph, letus move the feed-line entry point farther away fromthe load, from L = 30° to L = 60°. This increasesthe transformer length to 60 degrees and reducesthe stub length to 26.6 degrees. From observing theradial line extending from the L = 60° point (whereθ = -120°) through point B, where the SWR = 3.0and the R = 0.43 circles intersect on the VectorGraph, we see that the corresponding movementalong the SWR circle results in a change in theresistance component at the feed point from R =1.0 to a new resistance R = 0.43.

Recalling from the earlier statement that thecomplementary mismatch relation holds constantwherever the feed line is attached, a feed linehaving a characteristic impedance ZC = 0.43 will beperfectly matched when attached at the L = 60°point. The load-mismatch voltage reflectioncoefficient is now read as ρ E = 0.5 ∠ - 120° at pointB with the same magnitude as before, but with alarger phase angle because we are farther from theload. And applying the complementary-mismatchprinciple we see that the voltage reflection from thestub mismatch becomes ρ E = 0.5 ∠ + 120°, asread at point C. So we ask the question, how doesthis new combination produce a canceling wavehaving the same magnitude as before? We willrecall that previously, when the line-characteristicimpedances of sections F, S and T were each ZC =1.0 as in Figure 7B, the canceling reflection wasgenerated by the stub alone, because no line-junction mismatch existed between the feed line Fand the transformer T. But now that the impedanceof the feed line differs from the impedance of thetransformer, we have an additional discontinuity at

the feed point, which also generates a reflection.And the shorter (series) stub portion now generatesa reflection which is smaller than when all the linesections had a ZC = 1.0, the stub-reflectionmagnitude being reduced by the amount of thereflection presently being generated by the feed-line to transformer mismatch. Thus by thecomplementary mismatch principle the resultingcanceling wave still retains the correct magnitudeand phase to cancel the load-mismatch reflectedwave at this new feed point on the transformer.This canceling wave is evidently generated by thecombined discontinuities of both the differing lineimpedances at the junction, and of the stub with thecorrected length. We have thus matched a feed lineof ZC = 0.43 to a load of ZL = 3 + j0 through atransformer of ZC = 1.0.

Using this same line of reasoning we may,conversely, shorten the transformer section andchange the stub section according to the tangentrelation in Eq. 10 to obtain a match for feed lines ofhigher impedance. We need merely to position thefeed-line entry point where the resistancecomponent of impedance in the transformer T hasbeen transformed to the value of the feed-lineimpedance ZC that we wish to use and then adjustthe stub length to cancel the line reactance. Asexplained above, the resistance circle which isintersected by the SWR circle for a giventransformer length indicates the feed-pointresistance component. This is also the ZC value ofthe feed line which will be perfectly matched whenattached at the feed point. The data presented inTable 1, taken from points along the SWR = 3.0

circle on the Vector Graph. show a few selectedtransformer-length examples and pinpoint some ofthe pertinent information for clarity. Noticeespecially that the resistance component decreasesas the transformer length increases. It is interestingto discover that when the feed point goes beyondthe L = 45° position, the θ angles of the voltage-and current-reflection coefficients pass through 90degrees from opposite directions, respectively. Theresult of this is that their respective resultants shift180 degrees. Thus the resultant reflection-coefficient angles interchange, the voltage-coefficient resultant angle θ now becoming 180degrees and the current angle becoming 0 degrees.This means that the effective reflecting terminationat the match point shifts from an open circuit to ashort circuit when the feed line is attached morethan 45 degrees away from the Emax position at L =0° on the transformer.

Consider now the effect of increasing thelength of the transformer section still further, untilthe reactance component of the line impedancedisappears by itself without requiring a stub tocancel it. The Vector Graph shows this condition tooccur at L = 90°. At this point the load-mismatchreflected voltage wave is exactly 180 degrees outof phase with the incident wave and therefore noreactance component is developed here. Theresistance component of the line impedance at thispoint is 0.33 x Z0 on the chart. Based on our presentreasoning, we may at this point connect a feed lineF having an impedance ZC = 0.33 (see

Table I - Matching characteristics with various transformer- and stub-section lengths.

Angle of Reflection Coefficient

Voltage CurrentTransformer Stub Resistance Stub Load Stub & Load Stub & linelength length component reactance mismatch line mis- mismatch mismatch

L° SL° R jX θ° match θ° θ° θ°0 0 3.0 0.0 0 0 180 180

10 48.2 2.42 +1.12 -20 +20 +160 -16022.5 52.9 1.38 +1.32 -45 +45 +135 -13530 49.0 1.00 +1.15 -60 +60 +120 -12045 38.7 0.60 +0.80 -90 +90 +90 -9052 33.0 0.50 +0.65 -104 +104 +76 -7660 26.6 0.43 +0.50 -120 +120 +60 -6067.5 19.8 0.38 +0.36 -135 +135 +45 -4590 0 0.333 0.0 180 180 0 0

Fig. 7D) and obtain a perfect match. No reflectionswill appear on the 0.33-ohm feed line. How come?Again, because of the canceling reflections in thetransformer section! Note the present length of thetransformer section -- 90 degrees, or a quarterwavelength. The transformer section alone is nowusing the entire length of the line section, and thestub section has disappeared. Simply by movingthe feed point along the transformer to the pointwhere the line reactance vanishes, the resistancecomponent becomes ZC /SWR = 1/3, and we slipsmoothly from the stub form into the series quarter-wave transformer form of matching. (See Table 1.)Remember, the characteristic impedance of thetransformer is still ZC = 1.0, which has become thegeometric mean between the input and outputimpedances that it is matching 033 3 0. .x = 1.0.Looking from inside the transformer, theimpedance level at the input terminals is steppeddown 3:1 (giving us short-circuit reflectionbehavior), just as the output impedance (load) isstepped up 1:3 (for an open-circuit behavior at theoutput terminals). It is therefore evident that a λ/4-transformer section of line having a ZC equal to thegeometric mean of its two end-terminalimpedances has equal mismatches at both ends, andthus produces reflections of equal magnitude atboth ends. These reflections from each end canceleach other out at the feed-line transformer-inputjunction, because waves reflected at the outputmismatch return to the input junction exactly 180

degrees out of phase with the waves reflected at theinput mismatch. This is because the load-mismatchreflected wave has traveled 90 degrees from theinput point to the load mismatch, and an additional90 degrees in returning to the input. To clarifyfurther what is happening here, we recall thatpreviously, when the feedline F and transformer Twere of equal impedance ZC (Fig. 7B), thecanceling reflected wave was generated entirely bythe stub mismatch. In the present case where thestub length is zero, the canceling reflected wave isgenerated entirely by the 3:1 feed-line transformer-junction mismatch. The voltage reflectioncoefficient angle of this feed-line junction-mismatch reflected wave is θ = 0° referenced to thefeed-line ZC because the transformer ZC is threetimes greater. However, after both the load- andinput-junction-mismatch reflected waves havejoined to cancel one another, the input-junctionreflection no longer travels toward the generator,but is rereflected into the transformer toward theload in the same manner as with the previous stub-reflected wave. Therefore, referenced from withinthe transformer, the reflection coefficient angle ofthe junction reflection is θ = 180°, as indicated inTable 1. In a later section we will see why the λ/4series transformer is an impedance inverter for anycomplex terminating load, and is not restricted topurely resistive loads.

Part 5 - Low SWR for the Wrong Reasons

IN PART 1 of this series of articles thestatement was made that misconceptionsconcerning SWR and reflections are rampantamong amateurs, both in print and on the air. So toreiterate further, this series has been written withone primary goal in view -- to identify some of themisconceptions and to provide correct answers inthe hope of clarifying some of the confusionresulting from the misconceptions. Oneoutstanding area of confusion concerns the natureof reflected power and how it is accounted for inthe circuit. In short, is it real or is it fictitious, andwhere does it go? Now the nature of reflectedpower was discussed in Part 3, where it was shownwhy reflected power is real power. And in Part 4we delved into the question of where the reflectedpower goes, as the role of reflections in conjugatematching was discussed. There the stub form ofmatching was used to illustrate the wave actionwhich accomplishes the matching function andwhich also derives the total incident power fromthe combined source and reflected power. We willrecall that learning of this wave action strippedaway the mystery of how a mismatched load canabsorb all of the power delivered by the source.We learned this as we saw how the reflected power

adds to the source power at the conjugate matchpoint so that the reflected power can be subtractedfrom the total, enlarged, incident power at themismatch point to leave a net power in the loadequal to the source power.

Now that we have established thisrelationship between the source, reflected, andincident powers in terms of the wave mechanics ofthe conjugate match, we have the necessarybackground and tools for identifying some of theimproper usage of SWR, and for clarifying ingreater detail the reasons for the misunderstandingthat still prevails concerning what happens to thepower reflected from an antenna that is mismatchedto its feed line. Further clarification of themisconceptions will enhance the appreciation ofthe mismatched feed line as simply an impedance-transforming device, particularly as we seesomewhat later how the Transmatch type of feed-line matching network and the pi-network tankcircuit of the transmitter perform the conjugatematching function in the same manner as the stub.Additional perspective in relating the discussion topractical feed-line operation will be gained as someof the thoughts presented in Parts 1 and 2 of thisseries are now expanded.

If it appeared to some readers that theimportance of SWR was overly minimized ordowngraded in the treatment accorded it in Part 1,it was not so intended. The intent there was tofocus attention on the importance of understandingthe subject of reflection and SWR correctly and insuch depth that we may retain complete controlover them in our antenna system designengineering. Thus, instead of letting SWR becomeking to take control and deprive us of a breadth andflexibility, we may use SWR in the system designchoices in ways which many are unaware exist.

How many of us have acquiesced to theKing in pruning an 80-meter dipole, with greatpains to obtain the best possible match to a half-wavelength feed line at a specific frequency, butcannot operate more than a few kHz from thatfrequency without fear of the King's apparent direconsequences? But how many are aware that KingSWR can be outwitted and his consequencesaverted without pruning either the dipole or the

feed line? And how many have been aware that thematching operation can be performed at thetransmitter end of the line at any frequency withinthe entire 75-80 meter band without suffering anysignificant loss in power in spite of the SWRremaining on the feed line? Although it contradictsthe word published in many articles during the pasttwo decades, this revelation is true, and isindicative of the flexibility or freedom that really isavailable in our choice of antenna systemsdesigned for all the hf bands, simply by having abetter understanding of SWR and reflection.

Valid Reasons for Low SWR

There are good and valid reasons for beingconcerned with SWR and reflection, from both theamateur and commercial viewpoints -- with thisthere can be no argument. As we well know, thesereasons are concerned basically with voltagebreakdown and power-handling capability,efficiency and losses, and with line-inputimpedance as it relates to transmitter outputcoupling. In amateur practice, power-handlingcapability and voltage breakdown don't becomeserious problems unless we try to shove the legallimit of power through RG-58/U or RG-59/U at ahigh SWR. Losses and efficiency concern us, but toa much smaller degree than is generally realized,and for a different reason than many are aware, aswill be shown very shortly.

The chief reason why the amateur should beconcerned (but not alarmed) with SWR is in itsrelation to line-input impedance and transmittercoupling. This will be discussed in great detail in alater section. There we will see how to tameimpedance and coupling for any reasonable valueof SWR, and in that discussion the relativeunimportance of having a resonant antenna willalso become evident. But it is of great importancethat we first clarify some of the prevalentmisunderstandings of SWR and reflected power,because they are causing many amateurs to strivefor a low SWR for wrong, invalid reasons, andoften needlessly. Probably the most serious andwidespread misconception concerning SWRprevailing throughout the amateur fraternity is theerroneous notion that there is a direct one-for-onerelationship between reduction in reflected powerand a resulting, increase in radiated power. In otherwords, every decreased watt of reflected power is

thought to provide an additional watt of increasedoutput. Not So, but the tremendous number ofamateurs who have been misled to believe thisinvalid, unscientific, and untenable premise issimply unbelievable.

Another related concept, popular, but alsoerroneous, is that when terminated in a mismatch,the coaxial feed line becomes part of the radiator,causing radiation from the feed line due to thestanding wave (ref. Part 2 of this series, statement18). This is untrue because the line voltages andcurrents, and the standing waves resulting from themismatch, are entirely contained in and betweenthe outer and inner conductors, inside the coax. Nostanding wave develops on the outside because ofmismatch. However, feed line radiation may resultfrom standing waves on the outside of the coaxcaused by current unbalance if a balanced dipole isfed with coax and no balun is used. This feed-lineradiation may or may not be of any consequence,but the topic is covered well by McCoy (ref. 45).

Misunderstanding of how the benefitsaccrue, from a low SWR, and of just how littlebenefit is obtained, is driving many of us to attainSWR values far lower than where the benefitscontinue to be significant in relation to the effortsexpended to attain them. It is for this reason thatwe often set an unrealistically low limit on SWRthat needlessly restricts the operating bandwidth, orrange of usable frequencies on either side of theantenna resonant frequency, to a far more limitedrange than is necessary. In rectifying amisunderstanding such as this, it often helps tolearn first how the misunderstanding originated.

"Impedance" Bridges

One aspect of the misunderstanding hasbeen created to a large extent by narrow and oftenerroneous interpretations of matching principlesfound in various instructions for instruments suchas noise bridges2 3 and the antenna-scope fordetermining the terminal "impedance" of anantenna. Contrary to what is stated in some of theinstructions, these devices cannot measureimpedance -- they can measure resistance only --and then only in the absence of reactance.(Suggestion: Look up and compare the definitionsof impedance and resistance; the term impedance isoften misused when the correct term should beresistance. The reader is also invited to see ref. 46).

Consequently, in using these devices we have beencoerced into finding only the resistance componentof the antenna terminal impedance, and only at theresonant frequency of the antenna, because this isthe only frequency where the impedance has zeroreactance, or R + j0.

In following this tack, erroneous emphasishas been given to requiring the antenna radiatoritself to be resonant, thus nurturing themisconception that it needs to be resonant toradiate all the power being supplied to it.24, 25 Thus,many have been misled to believe that the antennajust won't perform properly at any frequencyexcept the resonant frequency. (See Part 2 of thisseries, p. 21, statements 5, 6 and 7 and refs. 20, 21,and 24.) In addition, emphasis on the furthernecessity for obtaining a resistance componentreading equal to the line impedance ZC has in manycases caused us to go to extreme lengths, such asadjusting the antenna height above ground in smallincrements to achieve that exact resistance readingin quest of the perfect 1.0 match.26 (See Part 2 ofthis series, p.22, statement 15.) Adjusting heightsin large increments to obtain control of radiation inthe vertical plane is realistic. But controllingradiation resistance by adjusting the height isneither necessary, realistic, nor practical, becausethe efficiency thought to be gained through thisaction is illusory.

The truth of this will become evidentsomewhat later as we see why there is nojustification whatever for expending any matchingeffort at the load to improve a mismatch of 2:1 orless, simply to remove the standing wave with theexpectation of improving efficiency. Furthermore,because of the reactance that appears as we departfrom the resonant frequency, the sacred butoverrated perfect match found at some carefullyadjusted height can be obtained at only onefrequency without retrimming the radiator length,thus continuing the vicious cycle. However, thewidespread practice of this philosophy in antenna-system operation has completely conditioned us tothink only in terms of using a λ/2 transmission linewith no reflection, and to obtain its perfect 50-ohmnonreactive input impedance by operating only atthe resonant frequency. So we have, in effect, beendeterred from learning of the real effect ofreactance in antenna impedance, and of how theline transforms any terminating impedance in astraightforward and predictable manner.

In becoming so conditioned, many of ushave forgotten that we can obtain the desired 50-ohm nonreactive input impedance from the line-transformed antenna impedance with a simple line-input matching network in the shack, often moreeasily than it can be obtained at the antenna. Infact, in some transmitters the impedance seen bythe transmitter at the line input for SWR values of2:1 or higher can be matched for optimum loadingby adjustment of the transmitter tank circuit itself.If a transmitter does not contain sufficientmatching range, a separate line-matching networkbetween the transmitter and line input offers a morejudicious matching arrangement than playinggames out at the antenna. We will see why thereare many situations where this same matchingapproach should be considered when the loadmismatch yields SWRs of even 5:1 or higher, asone departs from the self-resonant frequency of theradiator (ref. 24).

One further misconception exists that hasalso resulted in needless and unwarranted relianceon the λ/2 feed line to repeat the resonant antennaresistance at the transmitter. This one concerns theeffect of line-input reactance on tank-circuitresonance when the line with reflections is feddirectly by the pi network. Consider a tank circuitwhich is first loaded and tuned to resonance with aresistive load, and then when the load is changed toone containing reactance. If the tank componentshave sufficient retuning range to compensate forthe reflected reactance and return the circuit toresonance at the proper load level, all is well; thetubes still see a proper resistive load as before. Themisconception about this point has been generatedby some writers who apparently don't understandresonant circuits, for they proclaim that theretuning introduces reactance that detunes thecircuit, causing improper loading, and increasesplate current and dissipation. Not so -- much moredetail on this point will appear in a subsequent part.

Low SWR for the Wrong Reasons

We have discussed "low SWR for thewrong reason," as practiced (often unwittingly) inusing the perfectly matched antenna operated onlyat the self-resonant frequency of the radiatingelement. But another wrong reason for desiring alow SWR is interpreting feed-line SWR as the sole

criterion for indicating the quality of an antenna'sradiating performance across a band of frequencies,with low SWR across the band getting the ravesand high SWR getting the boos. This is a definitemisuse of SWR, because there are cases where thelow and high SWRs occur in just the oppositerelation, with respect to indicating antennaefficiency over a given bandwidth, for reasonswhich will be explained shortly. As a result of thismisuse of SWR, good antennas are too frequentlyrejected as "bad" because the feed-line SWRswings relatively high, and poor antennas areaccepted as "good" when the SWR remainsrelatively low.

In most cases the use of feed-line SWRalone to indicate antenna efficiency is completelyinvalid, because SWR indicates only the degree ofmismatch, not efficiency. However, we will seepresently how a relative change in SWR, to a valueeither lower or higher than a previous value knownto be correct in a given antenna system, canindicate that a change has occurred somewhere inthe system. That change may affect its radiatingefficiency. The popular vertical antenna havingfrom two to four ground radials (an insufficientnumber for efficient operation), or perhaps havingonly a buried water pipe or a driven rod for aground terminal, is one case where lower-than-normal SWR obtained over a frequency rangeindicates a poor quality of radiating efficiency,rather than a good one. But conversely, improvingthe ground system by adding a sufficient numberofradials can increase the radiating efficiency tonearly 100 percent, and this improvement will beaccompanied by a significant increase in SWRreadings over the same frequency range to highervalues, which are the normal or expected values.

With an adequate ground system, the SWRis predictable over the frequency range, because aload impedance of any specific R + jX value yieldsan exact SWR on a given feed line, and because weknow approximately what the antenna impedanceshould be at whatever frequency we may wish touse (refs. 4 7, 48, 49, and 58, p. 3-1). But when theground system is inadequate there is an unknownground-loss resistance added to the known antennaimpedance, which changes the SWR to somelower, unpredictable value. Yet, without beingaware of these facts, we often tend to be happier inthe discovery of an unsubstantiated low SWR thanwe do in determining whether we have SWR

values that should be obtained with the existingconfiguration. This is a very important concept thatrequires a clear understanding if we are to avoidmisinterpretation of SWR data in our effort tooptimize radiated power.

It will help in understanding this concept ifwe have a clear physical picture of how the ground-loss resistance develops. It appears that we havestill another misconception here, this oneconcerning the current and field behavior in thevertical-over-ground antenna system. Most of usknow that conventional grounding techniques usedfor lightning protection, such as rods or pipesdriven deeply into the ground, provide an excellentlow-resistance current path for the lightningcurrent. Many are unaware, however, that thesetechniques are totally inadequate for conducting theentirely different pattern of current flow of thevertical antenna system.

Vertical Radiator over Earth

Let us digress a moment for a brief lookinto the field and current behavior of the verticalantenna system, to see what type of ground systemit takes to meet the current-pattern requirements.Consider a base-fed vertical antenna; one terminalof a generator is connected to the base of thevertical radiator and the other generator terminal isconnected to ground, just below the base of theradiator. During the half cycle in which theconduction current in the antenna radiator flowsupward, all the current returns to ground throughdisplacement currents, which follow the lines offorce in the rf electric field through the radiator-to-ground capacitance. See Fig. 8. The electric fieldsurrounding the antenna, which excites thedisplacement currents, fills the entire volume ofspace surrounding the antenna in the shape of anoblate or somewhat squashed hemisphere. Thishemisphere intersects the ground to form animaginary circle having a radius of slightly over0.4λ for radiators of λ/4 in physical height. (Theradius decreases as the physical height of theradiator decreases.) The displacement currentsenter the ground everywhere over the entire surfacewithin the circle and then flow back radially toreach the grounded generator terminal. Althoughsome of the current penetrates somewhat moredeeply, most of the flow at frequencies above 3

MHz is restricted by skin effect to the upper fewinches of the ground.

Now a ground system comprising only asimple water pipe or a driven rod or two is simply aterminal -- the ground-feed terminal of the antennasystem. So all the returning currents must flowentirely through the poorly conducting ground fromall directions everywhere within the circle to reachthe terminal. This ground system is often measuredto have an "acceptably low" resistance at dc (whichmay be satisfactory for lightning protection), but itinjects a series loss resistance in the antenna circuitat rf. The rf resistance often exceeds the radiationresistance of the antenna itself! Adding two to fourwire radials to the system will provide goodconductivity toward the ground terminal for thecurrents which reach those radials, but only a tinyamount of the total current entering the surfaceinside the circle is intercepted by the radials. Thus,all the remaining currents still flow only throughthe lossy ground, and the result is that we still havea high loss resistance.

Now if a sufficient number of equallyspaced radials (90 to 100) extending out to 0.4λ arepresent to intercept all the currents, all thereturning displacement currents find highlyconductive paths everywhere within the circle,which lead the currents through negligible lossresistance directly back to the ground terminal ofthe generator. This can be visualized by examiningFig. 8. Currents which do enter the ground betweenthe closely spaced radials quickly diffract to aradial wire, and thus travel only a short distancethrough lossy earth before reaching a goodconductive path. Thus, with sufficient radials, wehave a nearly perfect ground system which addsonly a negligible amount of resistance to the trueantenna impedance measurable between theradiator base and ground terminals (refs. 20; 50; 51and 57, pp. 115-124). From this we can see whythe lightning-type ground system, although inprevalent use, is unsatisfactory for an efficientantenna system (ref. 57, p. 82).

Now we are not suggesting that λ/4antennas with less than ideal ground systemsshould not be used, nor that fair results cannot beobtained without their use. But the differencebetween no or few radials (3 or 4), compared to100, can amount to over 3 dB. This is far in excessof the loss resulting from an SWR of 4:1 or 5:1 onthe average coaxial feed line used by amateurs. The

Fig. 8 - The hemisphere of current which flows as aresult of capacitance of a λ/4 vertical radiator to the earth or aradial system. At frequencies above 3 MHz, rf currents flowprimarily in the top few inches of soil, as explained in thetext. Ground rods are of little value at these frequencies, andspikes or large nails are sufficient to secure the outside end ofeach radial wire. With a sufficient number of radials, annularwires inter-connecting the radials offer no improvement inantenna efficiency, as the current path is radial in nature.

point being emphasized here is that the value ofground resistance is unknown and unpredictable insystems using less than an adequate number ofradials. This makes the resulting SWR readingsunpredictable and therefore useless for the purposeof evaluating the absolute quality of the system,unless some means is available for determiningwhat the change in SWR would be if the lossresistance could be switched in or out.

In practical amateur installations, theground resistance will be sufficiently low if only 40to 50 radials are used with a λ/4 radiator. The smallimprovement in radiated power for the addition ofstill another 40 or 50 radials with the λ/4 radiatorwill probably not justify the extra cost and effort.However, if a short vertical antenna (from λ/8 orless to λ/4) is contemplated, it should beremembered that the radiation and terminalresistances decrease as the radiator is shortened.The ground resistance now becomes a larger part ofthe total resistance, decreasing the efficiency. Thusthe ground resistance should be kept as low aspossible for the full capability of the short antennato be realized (refs. 51; 56; 57, pp. 18-29). There ispractically no difference between the radiationcapabilities of the λ/4 antenna and a radiator evenshorter than λ/8, except for the effect of groundresistance and the loss in the resistance of the coilused to cancel the capacitive reactance in theterminal impedance of the shortened antenna. Theprofessional literature is replete with referencesconfirming this point (refs. 20, 24, 52, and 53).

Resistive Losses and SWR

In this section we will see how anyadditional resistive losses that are separable fromthe true antenna load impedance affect the true loadSWR. By separable are meant such losses asground-loss resistance, corroded connectors andother poor connections, cold-solder joints, and soon. These all contribute loss resistances that we cancontrol or reduce. In contrast is the resistivecomponent of the antenna terminal impedance,which comprises both the radiation resistance andthe inherent conductor-loss resistance in theradiating element. In most cases the conductor-lossresistance in practical radiating elements isnegligible, unless excessively small wire is used.

There are several useful relationshipsbetween load impedance Z = R + jX, lineimpedance, Z C, and SWR. For example, it is wellknown that when the load impedance is a pureresistance R, equal to the line impedance Z C, thereflection coefficient ρ is zero, and the standing-wave ratio is thus one to one. But the reflection isno longer zero and the SWR becomes equal to theratio R/ ZC when the resistance is larger than ZC, orZC /R when the resistance is smaller than Z C. It isalso well known that ρ and SWR increase with theaddition of any reactance component in the loadimpedance that increases the total reactance,whatever the resistive component may be (see Eq.1, Part 3 of this series). And as noted previously,any combination of R + jX yields an exact value ofSWR, when terminating a line of given impedanceZC. We also know that the reactance, X, appearingin the impedance at the terminals of an antennacontributes more to the rise in SWR at frequenciesaway from the antenna resonant frequency thandoes the change in resistance. This is because thereactance changes more rapidly than the resistanceduring the change in frequency (ref. 58, p. 3-1).

There is, however, an interestingrelationship which is not generally well knownbetween the resistance and reactance componentsof a load impedance. This relationship sheds lighton how the two components affect mismatchreflection and SWR, and also explains why theunknown ground resistance and other lossesmentioned above reduce the usefulness of SWRreadings. When reactance is present in the loadimpedance, the minimum possible SWR occurswhen the resistance R is greater than ZC. The value

of the resistance that yields the lowest SWR incombination with a given value of reactance in theload (which we will call the minimum-SWRresistance) is dependent solely on the reactancepresent. This value may be obtained from therelationship:

r x= +2 1 (Eq. 11)

where r is the minimum-SWR resistance and x isthe reactance present in the load, with both valuesnormalized to the system ZC. (See footnote 22, Part4 of this series regarding normalized impedances.Also see Feedback, p. 46 of QST for Nov., 1973.)It can be seen from Eq. 11 that when x becomeszero, r = 1, for an SWR of 1:1, but it is interestingto know that the resulting SWR always equalsexactly the arithmetic sum of the minimum SWRresistance value r, and the reactance value, x. Thislatter relationship will help us to understand howunwanted loss resistance, separable from the trueantenna-load impedance, affects SWR. In the caseof the vertical radiator over earth, theseunpredictable losses change the SWR from apredictable value, based on known availableantenna impedance data (ref. 57, p. 82), to someunpredictable and usually lower value. A generalapplication of the relationship is presented in thefollowing statements. When the resistancecomponent of the true load impedance is lowerthan the minimum-SWR resistance, as determinedfor any reactance component also present in thetrue load, adding of resistance separate from thetrue load impedance will cause the SWR todecrease from the value obtained with the true load.This is true until the total resistance is equal to theminimum-SWR resistance. Further addition ofresistance will cause the SWR to rise again. Thesestatements apply especially to the vertical antennaof λ/4 heights or less in proving why groundresistance which reduces the efficiency alsoreduces the SWR. This is because the true antennaresistance component, R, is generally less than theimpedance, ZC, of normally used feed lines, whilethe minimum-SWR resistance R is always equal toor greater than ZC.

The effect of reactance in the antennaimpedance raises an additional factor of importancein understanding the relationship between SWRvalues and antenna performance. As stated earlier,the rate at which SWR rises as the operating

frequency departs from the resonant frequency ofthe antenna depends on the resulting change in theimpedance at the antenna terminals, which in turnis dependent on the Q of the antenna. One factorthat has a primary influence on antenna Q is theamount of capacitance between the opposite halvesof the dipole. (Although it is more commonlycalled a monopole, a vertical antenna over groundcan also be considered as a dipole, because thelower half is simply the image of the upper half,with the opposite polarity.) This dipole capacitanceis determined by the ratio of the radiator length, L,to its diameter, D.

The L/D ratio (refs. 1; 58, p. 3-1) found inthe usual simple thin-wire dipole is very high,resulting in a low dipole capacitance and high Q,causing a rapid change in impedance, reflection,and SWR as frequency changes. This is why a thin-wire dipole is considered a narrow-band device.However, specific broad-banding steps may betaken to increase the dipole capacitance and thusreduce the Q and thereby the rate of change ofSWR. One such step, for example, is decreasingthe L/D ratio by using a multiwire cageconfiguration for each dipole half, or by fanningout multiple wires from the feed point. SWRcurves vs. frequency are valid here in comparingbandwidths obtained while experimenting withdifferent radiator configurations. However, anyseparable loss resistance must now be eitherminimized or held constant to prevent it fromintroducing unknown variables. Otherwise, theunknown variables can caused differing errors inthe SWR readings obtained with differentconfigurations, and thus render the results of theexperiment invalid. But unless actual broad-banding steps have been taken to reduce the Q, therate of change in SWR as frequency changes willnot differ dramatically between various types ofdipoles having roughly equivalent Q values. (Thesetypes include the so-called inverted V.) If adramatic difference is noted with no validbroadbanding steps taken, troubleshooting is calledfor to determine the cause. More than likely someunwanted loss resistance will be flushed out, if thisis the case.

The writer has seen SWR curves published,along with descriptions of quite simple antennas,where it would have been impossible for the SWRto remain as low as indicated over the frequencyrange shown; the antenna Q of the configuration

presented would have simply been too high. Twopossible explanations for this sort of contradictionare that (1) perhaps the readings were obtainedusing an inaccurate SWR indicator -- many read onthe very low side (refs. 40, 54), or (2) as suggestedabove, an unrecognized trouble existed somewherein the antenna system which was lowering the Q bymeans of a separable loss resistance. Yet thesearticles were published because the antennas theydescribed were purported to have "'improved SWRcharacteristics." How many times have you heardsomeone praise his newly hung skywire by simplytelling how low the SWR indicator reads across anentire band? It should now be clear that it cannot beemphasized too strongly that an unrecognized andunwanted loss resistance is an antenna system cancause a low SWR reading when it should not helow! So in a later section we will explore therelationship between antenna impedance and SWRin detail so that we may determine what is a properSWR for given conditions.

Reflected Power and SWR

Let us now return to the subject of why weworship low SWR for a wrong and invalid reason.As stated earlier, the misunderstanding of thisaspect of reflected power is based primarily on theprevalent, but erroneous, idea that any reduction inSWR or reflected power effected on a line feedingan antenna results in a direct one-for-one increasein radiated power. The erroneous reasoning in thisidea is in the assumption that if the power is beingreflected, it therefore cannot be absorbed in theload or radiated, and that the power which isreflected returns to be lost by dissipation in thetransmitter. The assumption is false on both points,for the truth is, because of the reflective conditionsin the circuitry used in coupling the transmitter tothe line, all power that enters the line is absorbedby the load (except that dissipated in the line itselfdue to attenuation). This is true even when the loadis not matched to the line impedance (ref. 55).Complete absorption in the mismatched load (andline) of all the power delivered by the transmitter isobtained, because the power reflected from themismatch is conserved and returned to the load byrereflection from the line-coupling or matchingcircuitry, in accordance with the principlesdiscussed in Part 4.

Let us consider a lossless line for amoment, in light of the above statement; here it isaxiomatic that if all power delivered to the line isalready being absorbed in the load (because nonecan be absorbed in a lossless line), a reduction ofthe reflected power cannot have any effectwhatever on the amount of power taken by theload. And obviously, there is no power left over tobe dissipated in the transmitter. Following thissame reasoning in a real line having attenuation, alllosses in power must be attributed to the basic I2Rand E2/R losses arising from the line resistance orattenuation. These losses are unavoidable, evenwhen the load is perfectly matched. The onlyadditional power losses which can be attributed toSWR or reflection occur because the same resistiveattenuation is encountered by the reflected powerwave as it travels along the line from the load tothe input. The amount of power lost in this manneris very small, indeed, at frequencies in the hf rangewhen good-quality low-loss line is used because,during its return to the input, the reflected powersuffers only the same rate of line-attenuation loss(in dB) as the incident power suffers in its forwardtravel toward the load. And as previously stated, allthe reflected power which arrives back at the inputnow becomes part of the incident power. Anotherway of explaining the relation between SWR andlost power is to recall from Parts 3 and 4 thatbecause the incident power is the sum of the sourceand reflected powers, the incident power is greaterthan the source power wherever the SWR is greaterthan 1.0. Thus for a given source power, theresistive losses are somewhat higher in the portionof line where the incident power is higher than thesource power, simply because the average linecurrent I and voltage E are higher in that portion.

So from this discussion concerningimproper usage of SWR we learn that from theviewpoint of efficiency, our concern for SWRinvolves only the loss due to line attenuation. Thuswe can tolerate a higher SWR when the attenuationis low, but when attenuation is high the SWR limitmust be lower for the same amount of additionalpower lost from SWR. The exact relation betweenSWR and the power loss caused by SWR fordifferent values of line attenuation is showngraphically in Fig. 1, Part 1, taken from the ARRLHandbook and the Antenna Book. From this figurewe can easily see that the amount of power actuallylost is in sharp contrast to the amount mistakenlyassumed to be lost in the improper concept ofSWR, where it is thought that a reduction in SWRor reflected power results in a direct equivalentdecrease in the amount of power lost in the system.

There is a great deal of irony behind thesevarious misunderstandings of reflections that haveengendered the wrong interpretation or usage ofSWR. The irony is that the correct reasons whySWR should be considered, as previously recited,are frequently overlooked in the wrong usage,while the basis so generally accepted in support ofthe wrong usage doesn't even exist in the couplingmethods used by amateurs to transfer power fromthe transmitter to the antenna. A part of this obtuselogic originated from the confusion among bothamateurs and engineers in the meaning of a "matched generator" -- to some it implies beingmatched in only one direction, and to others itmeans being matched in both directions. Intransmitter operation, where conjugate coupling isusually used to deliver optimum power to a loadthrough a line, the match is in one direction only --forward.

Part 6 - Low SWR for the Wrong Reasons (continued)

PART 5 OF THIS SERIES concluded withthe statement that in transmitter operation, whereconjugate coupling is usually used to deliveroptimum power to a load through a line, the matchis in one direction only -- forward. The generator(transmitter) is matched to the line, but, lookingback into the generator coupling circuitry during alltimes that the generator is actively supplying powerthrough the conjugate coupling to the line, the lineis totally mismatched. The conjugate relationshipmay be demonstrated by making impedancemeasurements in either direction from any point onthe line. These measurements will show animpedance R + jX looking in one direction, and theequal but opposite-sign impedance R - jX in theopposite direction. (The net reactance of zeroobtained from these two impedances proves thesystem is resonant!) But these measurementscannot be performed while the generator is active;it must be turned off and replaced with a passiveimpedance equal to its optimum load impedance. Inthis case the impedance now terminating thegenerator end of the line will be seen as adissipative load while measuring impedance in thegenerator direction. (See footnote 5, Part 2 of thisseries.)

The fact that dissipation occurs in theimpedance which replaces the generator impedanceduring these measurements is largely responsiblefor the erroneous inference that power reflected inthe generator direction is also dissipated in asimilar manner in the generator impedance.However, when the generator is active, its internalimpedance is never seen as a load for powerreflected from a mismatched load terminating theline because of the interaction between the sourcewave, the load-reflected wave, and the cancelingwave, as described in detail in Part 4. The fine isthus totally mismatched looking in the generatordirection.

In laboratory work, on the other hand, thegenerator is usually matched in both directions.Here the generator is isolated from the line with aresistive pad, or attenuator, having an insertion lossof about 20 dB, and having the same impedance asthe generator output and the line ZC. In this case thegenerator sees a match looking into the line-

terminated pad, and the line also sees a matchlooking back into the pad. This is because the padabsorbs and dissipates both forward and reflectedpower like a lossy line, so that only about 1/100 ofthe source power reaches the load, and any powerreflected from a mismatched load is also dissipatedto 1/100 of its original value during its return to thegenerator. As a result, the reflected power reachingthe source is about 40 dB below, or 1/10,000 of thepower delivered by the generator, when the load isa totally reflecting short- or open-circuittermination, and even less with practicalterminations that are dissipative. This amount ofreflected power reaching the generator is negligiblefrom the viewpoint of adding to the source powerand modifying the line-input impedance. Thus, forall practical purposes, the pad appears to thegenerator as either an infinitely long line, or a linehaving a perfect R = ZC termination, and both aconstant generator loading and constant incidentvoltage are maintained as line loading is changed,to satisfy laboratory requirements. (See Part 1 ofthis series, p. 37, last paragraph, and ref. 19, pp. 7and 48.) Thus, it is understandable that confusionbetween these two forms of matching can beresponsible for misleading us into thinking thatreflected power in the transmitter case is dissipatedand lost on return to the source.

Reflected Versus "Lost" Power

This erroneous concept of reflected poweris widespread, having been nurtured on the air for along time, and supported in print in so manypublished articles it would be impossible to countthem. Two such articles, one by K8ZVF, and theother, an SWR-indicator review by W2AEF, areespecially pertinent, because they contain explicitstatements supporting the erroneous concept,whereas statements in many other articles onlysupport the error implicitly.

Let us now make a further analysis of thereflection mechanics involved in generatormatching, in which two important ingredients thathave been overlooked for a long time will berevealed. In so doing, we will see not only whystatements concerning lost power published in the

two articles mentioned above are incorrect, but alsowhy it was so easy for these ingredients to beoverlooked early in the amateur use of coaxialtransmission line, with the result that many havebeen misled into seeking low SWR for the wrongreason.

Assume a lossless transmission line havinga perfectly matched load termination. Assume alsoa matched condition between the generator ortransmitter and the line characteristic impedance,ZC. With these conditions there is no reflectedpower in the line and therefore no reflection loss.The generator delivers what is defined as themaximum-available matched power, and the loadabsorbs all the power delivered. If the loadtermination is now changed, creating a mismatchbetween the line impedance ZC and the terminatingload, less power will be absorbed by the load. Theamount of the reduction in absorbed powerresulting from the change in load impedance is themeasure of the reflection loss. As the reflectedpower wave returns toward the generator it causesa change in the line impedance from ZC to Z = E/Iall along the line, as stated in Part 3, and as shownfor an SWR = 3.0 in Fig. 4. When the reflectedwave reaches the input terminals of the line thegenerator is presented with a change in line inputimpedance from the Zc value to some new valuedetermined by the E/I-vector relationship at theline-input terminals. This new impedance at theline input has exactly the same degree of mismatchto the line ZC as the terminating load that generatedthe reflection. Thus, the line is also nowmismatched to the generator in the same degree,and in this condition the generator willautomatically make less power available to the line.

The reduction of power delivered to the lineis exactly the same amount as the power reflectedat the load. In other words, the reflection loss at theload can be referred back along the line to thegenerator. Thus, reflection loss is simply a non-dissipative type of loss representing only theunavailability of power to the load due to thegenerator's making less power available to the lineas a result of the mismatch of impedances causedoriginally by the mismatch at the load terminatingthe line. (That reflection loss represents only theunavailability of power to the load will becomeevident as it is now shown that the load absorbs allthe power the generator makes available to theline.) On reaching the generator terminals and

causing the mismatch to the generator, the reflectedpower is totally rereflected toward the load, addingto the source power exactly the same amount as thereduction in power made available by thegenerator. Since incident power equals sourcepower plus reflected power, the incident powerreaching the mismatched load remains the same asbefore the generator made less power available.The reflection loss therefore now equals thereduction in generator power.

If a conjugate match is now providedanywhere along the line, even at the inputterminals, the reflected-power wave is preventedfrom traveling past the match point toward thegenerator, as explained in Part 4. Thus, the lineimpedance between the match point and thegenerator is now unaffected by the reflected wave,and remains at its ZC value. Also, the generator nolonger sees a mismatch, and again delivers itsmaximum-available matched power to the line. Theconjugate match has thus provided a negativereflection, commonly called "reflection gain,"which exactly equals and cancels the reflectionloss. But it has also been shown that all the powerdelivered by the generator is absorbed in the loadin either case -- with, or without the reflection gain-- the generator simply made less power availablebefore the reflection gain restored the matchedcondition between the generator and the line. (Ref.19, p. 37; Part 2 para. 3, statement 9; Part 4, para.2.)

So we now ask the question, how does thissituation relate to the K8ZVF nomograph, wherereflected power is stated to be "lost power," and tothe "useful power" table from the Knight SWRindicator review? (See footnotes 27 and 28.) It isthis: The nomograph simply converts SWR back toreflected power ρ2, which is what the SWRindicator actually measures but converts to SWRby means of its scale construction. As discussed inPart 3, ρ2 is the measure of reflection loss or powerreflected, which, as stated above, equals thereduction in power made available from thetransmitter, calculated directly from the mismatchbetween the line ZC and the load impedance ZL. Thereflected power is the square of the voltage- orcurrent-reflection coefficient, p, from Eq. 1, Part 3(see also Fig. 4), but remember further that it is a

Fig. 9 - Reflection loss versus SWR and matched-line loss of rf transmission lines. Total attenuation in a line operatingwith SWR may be determined from the dB scale at the right of the chart. The calibration scales at the left are discussedin the text. The a curves in the body of the chart represent the matched-line loss for a particular length of line at aparticular frequency. For example, the following types and lengths of line would exhibit the a attenuation factorsindicated. Each of these examples is for a frequency of 4 MHz: a = 0.03 dB - 100' of No. 12 open-wire line; a = 0.064dB 20' of RG-8/U; a = 0.1 dB - 100' of Amphenol Twin-Lead, No. 214-022; a = 0.2 dB - 62-1/2' of RG-8/U1; a = 0.32dB - 50' of RG-59/U, 100' of RG-8/U, or 200' of RG-17/U; a = 0.5 dB - 87" of RG-59/U or 175' of RG-8/U; a = 0.64 dB- 100' of RG-59/U or 200' of RG-8/U; a = 1.0 dB - 119' of RG-58/U, 350' of RG-8/U, or 700' of RG-17/U. The curvesare plots of the following expressions:

INCIDENT, OR FORWARD POWER POWER POWER(multiply by source power delivered) REFLECTED ABSORBED

POWER AT POWER AT LOADCONJUGATE (after line attenuation)MATCH POINT

11 2 4− −ρ e a

ee

a

a

−

−−

2

2 41 ρ_

ρρ

2 2

2 41e

e

a

a

−

−−=

( )1

1

2 2

2 4

−−

−

−

ρρ

e

e

a

a (Eq. 12)

with lossless

line (a = 0)1

1 2− ρ1

1 2− ρ_

ρρ

2

21−= 1 (Eq. 13)

where ρρρρ = magnitude of voltage-reflection coefficient (see Part 3, para. 1)

a = line attenuation in nepers (= dB

8 686.)

e = 2.71828, the base of natural logarithms

POWER ABSORBED = ( )

( )1

1

2

2

−−

×ρρ

less one - way line attenuation

less two - way line attenuation source power

nondissipative power, because it all eventuallyreaches the load, as explained in Parts 4 and 5.

The tabularized data in the SWR-indicatorreview article correctly lists percentage of reflectedpower ρ2 for corresponding values of SWR. But the"useful-power" column is incorrectly labeled and istherefore misleading, because it is actually listingpercentage values of (1 - ρ2), which is the portionof the maximum-available matched power thetransmitter actually delivers, depending on thedegree of mismatch it sees. In other words, thiscolumn is simply specifying the amount of powerthe transmitter will deliver into the mismatch iffirst tuned to a line having a matched ZC load, andis then switched to the mismatched load withoutthe benefit of retuning or rematching to the newimpedance at the line input. But we do not operatein this manner -- we retune, thereby matching thetransmitter to the new load and consequentlyestablishing the reflection gain 1

112−

−

ρ which

completely cancels the reflection loss ρ2 and theeffect of the load mismatch; the transmitter nowreturns to delivering 100% of its matched availablepower to the line, whatever the SWR on the linemay be!

Thus, the two missing ingredients are:1 ) Understanding the concept of reflection

loss and reflection gain, and2) The discovery that the reflected power is

totally rereflected at the generator terminals, eitherwith or without the reflection gain.

It is now evident that the informationpresented by K8ZVF and W2AEF is not specifying"lost" power at all, but only the reflection loss theamount of power made unavailable by thetransmitter until the conjugate match provides thereflection gain which cancels the loss and permitsthe transmitter to deliver its maximum-availablematched power. And as stated on several previousoccasions, the conjugate match is automaticallyattained (sometimes unwittingly) either by propertuning of the transmitter tank circuit to the lineimpedance E/I, or (sometimes knowingly) by useof a line-matching network if the transmitter tanklacks sufficient range to obtain the match by itself.How the tank performs the conjugate match, andthe effects of undercoupling, overcoupling, andpossible reactive loading of the tank which canresult in the absence of the conjugate match will beexplained in detail later in this series.

Reflection Gain

Now refer to Fig. 9 (previous page). Thisfigure was developed to illustrate the reflection-gain concept, in order to emphasize the effect ofmisinterpreting reflection loss to be "lost" ordissipated power. The impact of this singlemisunderstanding of transmission-line principleshas been disastrous, because it is the principalcause of the prevalent low-VSWR "mania" (low-vis-war-ma'nya). It is the reason why so many of uswrongly believe that "getting the SWR down" isthe most important factor in "getting the power intothe antenna." We fail to realize that, whatever theSWR, with a low-loss feed line, the reflection gainhas canceled the effect of the load mismatch if thetransmitter can be made to tune and load properlyinto the line, and all the transmitter power isalready being taken by the antenna. And therefore,as explained previously in Part 5, we have alsobeen unaware that no more power of anysignificance reaches the antenna in achieving thelower SWR. We have also been unaware that lineattenuation is the key to whether the SWR level hasany practical impact on efficiency at all (ref. Part Iof this series, p. 38).

In Fig. 9 the heavy curve marked ρ2 and (1 -ρ2 ) is based on lossless-line conditions. It is also anexact replot of the K8ZVF "lost power"nomograph, and indicates both the reflected and so-called "useful power" columns from the reviewarticle. The curve represents reflected power, ρ2 ,versus SWR, if one reads downward from the topof the chart. The power made available by thetransmitter versus the mismatch it sees in terms ofSWR, (1 - ρ2), is found by reading upward from thebottom. Thus, in reading upward, the curverepresents the power being made available with thetransmitter tuned for a perfect ZC match, butactually looking into an uncorrected mismatch.However, as explained above, when the reflectionloss is canceled by the reflection gain of theconjugate match obtained by retuning thetransmitter, a new curve, α = 0.0 dB, which nowrepresents this matched condition, follows theheavy straight line across the top of the graph. Thisindicates that 100% of the power is being madeavailable, and is also taken by the load regardlessof the SWR value. Suddenly the "lost" power isfound!

As stated previously, power can be "lost" ina transmission line only through line attenuation -if the attenuation is zero, lost power is also zero, asshown along the α = 0.0 dB curve, Fig. 9. Themore attenuation, the more power is lost, as shownby the various loss curves marked α = 0.03 dB andso forth. Since no allowance for the attenuationfactor was made in either case in the materialreferenced in footnotes 27 and 28, we have,therefore, still another reason why the terms "lostpower" in one case and "useful power" in the otherare incorrect and misleading. (Ref. 55)

A later part of this series will deal in detailwith attenuation effects, and will show how toperform some of the pertinent calculations basedon the W2DU- derived equations associated withFig. 9. However, the loss curves in Fig. 9 are in aform practical for visualizing the correctrelationship between the losses actuallyencountered in different feed fines of variouslengths, values of attenuation, and values of SWR.The curves indicate the total loss encountered onthe line due to attenuation. The curves represent thecondition in which the transmitter is conjugatelymatched at the input to the line, and thereforesignify that the effect of the load mismatch iscanceled in each case. Each curve starts at the left,where the SWR is 1.0, thus indicating the actualattenuation encountered by a particular line whenterminated in a perfectly matched load. The lossvalue along each curve is seen to increaselogarithmically as SWR increases on the line due toincreasing mismatch of the line termination. Thus,the difference between the loss incurred with a 1.0-SWR termination compared to that for any othergiven SWR value on the same line indicates theamount of additional loss that will be incurred forthat SWR. Here is further graphical evidence thatwhen the line attenuation is low, the additional lossdue to reflection is surprisingly small, even whenthe SWR is quite high.

Examine the SWR region between 1:1 and2:1. Do you see enough difference in power levelon any of these lines to justify any effort inreducing a 2:1 SWR to any lower value whatever?Do you still think you'll get out better by squeezingthat SWR of 1.8 down to 1.2? A review of pages36 through 38 and page 40 of Part 1 (April QST,1973) is now both pertinent and appropriate toemphasize how the use of these concepts canbroaden our design flexibility. It is also suggested

that the reader check the efficiency values of boththe spacecraft feedline examples on page 37 andthe 80- and 40-meter dipole examples on page 40in the Fig. 9 graph. In the NAVSAT example,comparison of the 66% value of reflected-powerwith only 1.15 dB actual loss will be especiallyrevealing.

Fig. 10 provides additional line-attenuationdata which permit us to extend the use of the Fig. 9curves to other frequencies and lines. Fig. 10 mayalso be supplemented with further data available inthe ARRL Handbook and the Antenna Book. (refs.1 and 2).

Radiation Resistance

In Part 2 of this series, statement 26 says, ineffect, that no significant amount of power will besaved by using matching circuitry between the feedline and the antenna terminals of a mobile antennasystem in the 80- through 10-meter bands. Andstatement 27 goes on to say that in the absence ofsuch matching circuitry, more power is radiatedfrom center-loaded mobile antennas that have ahigh feed-line SWR at resonance than those thathave low SWR. The concepts involved in thosetwo statements are also rather widelymisunderstood, so this is an appropriate time toclarify both of these statements, because theseconcepts also fit the category of "low SWR for thewrong reasons."

It is well known that the radiation resistanceof the short mobile antenna is very low. And of allthe hf amateur bands, the radiation resistance is thelowest at 80 meters, because the electrical length ofthe radiating portion is the shortest on this band.Depending on the exact length of the antenna andother factors, the radiation resistance of the center-loaded antenna is approximately 1.0 ohm in thisband, as shown by Belrose (ref. 60). Thecapacitive-reactance component in the terminalimpedance of this short antenna, which ranges from-j3000 to -j3500 ohms in typical 80-meter models(also shown by Belrose and confirmed frommeasurements made by the author), is canceled bythe equal +j inductive reactance of the loading coil.

However, it is not well known that there aretwo other resistances which become important forconsideration in antennas of this type. Theseresistances, coil loss and ground loss, add to theradiation resistance to comprise the total resistive

component of the impedance appearing at theantenna terminals. Thus it is erroneously thoughtby many amateurs that the one-ohm radiationresistance alone comprises the entire antenna-terminal impedance, and also that a matchingdevice is thereby required at the antenna to matchwhat is thought would be about a 50:1 mismatch iffed directly with a 50-ohm feed line. Actually, lossresistance in the loading coil and any ground-lossresistance both add to the radiation resistance,causing the terminal resistance to be much higherthan is commonly realized, though still lower thanthe impedance ZC of normally used feed lines. Thusthe actual mismatch value is much lower than isusually appreciated.

Fig. 10 - Attenuation in decibels per hundred feet forvarious coaxial cables.

While there are some who recognize thatloading-coil loss appears as part of the totalterminal impedance, only a few are aware thatground-resistance loss exists, because, exceptingBelrose, most writers neglect to mention it orconsider it in their system analysis. For example,see Swafford, "Improved Coax Feed for LowFrequency Mobile Antennas," QST for Dec., 1951.The Mobile Handbook, p. 100, (CowanPublications, 1st Edition) not only fails torecognize the existence of ground resistance, butmistakes what is actually the combined ground andradiation resistances as radiation resistance alone.(In subtracting 6 ohms of loading-coil and whipresistance from the 14 ohms of measured terminalresistance, the 8-ohm difference is simply taken to

be radiation resistance, with no mention of anyground resistance.) Thus, the ground resistance,which cannot be ignored, is unknowingly andimproperly being included as a portion of theradiation factor in the efficiency expression, insteadof as a loss. This oversight might have beenavoided if an analysis had been made of theincrease in radiation resistance actually obtained byraising the loading coil from the base to the centerof the whip, because the 6 ohms plus a 1-ohmradiation resistance subtracted from the 14 ohmsmeasured leaves 7 ohms, which requires anexplanation of where the remaining 7 ohms ofresistance came from. From further study of theCowan text, it appears that an assumption of agreater amount of radiation from the coil than whatthis author considers possible may have been thereason why such a high value of radiationresistance was considered plausible. But whateverthe reason, we have been given unrealistically highradiation resistance (8 ohms) and efficiency values(8/14 = 57%) that are fundamentally impossible toobtain in the center-loaded mobile antenna, and thetrue values have been obscured. In other words, alarge portion of the power considered in the CowanMobile Handbook as being radiated is actuallydissipated as heat in the ground. Belrose shows aproper analysis (ref. 60), supported by this author'smeasurements.

Now we will see why it's practicallyimpossible even to obtain a mismatch of sufficientmagnitude that will require any matching circuitrybetween the feed line and a properly resonated,conventional, center-loaded mobile antenna for thepurpose of conserving any significant amount ofpower, the many remotely controlled luggagecompartment tuning and matching arrangementsnotwithstanding.

Loading-coil loss resistances range fromabout 8 ohms for the better commercially availablecoils to as high as 31 ohms, measured in poorercoils, depending on the Q and the self-resonantfrequency of the coil. Ground-loss resistanceencountered with conventional low-band mobileantenna installations ranges from about 5 ohms forlow, wet ground to around 12 ohms for high, dryground; average ground yields around 7 ohms. Theground-loss resistance in the mobile setup is lessthan that found in an antenna of a full λ/4 inphysical height with no radials, because the radiusof the circle where the minimum currents return to

the ground is shorter with the shorter antenna andtherefore the currents travel a shorter distancethrough lossy earth (ref. Part 5, Fig. 8). (Thecurrent-flow pattern in the mobile system is alsodescribed by Belrose.) Thus, the total terminalresistance is nowhere near one ohm but lies in therange from an absolute minimum of around 14ohms when using loading coils of high Q to as highas 44 or 45 ohms when higher ground loss and theuse of a poorer quality coil occur simultaneously.Referenced to a 50-ohm feed line, these resistancesresult in SWRs (at the resonant frequency) fromaround 3.5:1 (with the lower loss resistances) downto around 1.1:1 with the higher losses.

Thus, as strange as it may seem, the higherthe minimum SWR attainable at resonance, thegreater will be the power radiated (for the samepower delivered to the feed line by the transmitter).It will not seem so strange, however, when weconsider that the low radiation resistance of aroundone ohm is the only portion of the total resistancethat contributes to radiation -- and it is constant --fixed by the radiator length. So by making the lossresistance lower through the use of a higher-Qloading coil, less power is dissipated in heat,leaving more to be radiated. Conversely, if thelower Q coil is used simply to achieve a lowerSWR, less power is radiated because more is nowbeing spent in heating the coil.29 However, whenusing the higher-Q, lower-loss loading coil, eventhough its lower loss resistance results in a largerload mismatch and higher feed-line SWR, theincrease in radiated power is still proportional tothe decrease in total resistance. Any additional lossfrom the higher SWR is so small it can beneglected, because line attenuation in the short feedlines used in mobiles is extremely low. Remember,line attenuation is the only cause of power loss inthe feed line, regardless of the SWR level.

In Fig. 9 the loss curve α = 0.064 dBrepresents the loss characteristics of a typicalmobile feed line -- 20 feet of RG-8/U at 4.0 MHz.The curve shows a matched-line attenuation of0.064 dB, plus an additional loss of 0.056 dB atSWR = 3.5, for a total loss of 0.12 dB. When thiscondition is accompanied by the 1:14 ratio ofradiation resistance to total terminal resistance anda transmitter power of 100 watts, the difference inpower radiated between matching at the antennaterminals and leaving the 3.5:1 SWR on the line

and matching at the line input amounts to less than0.1 watt!

It is also of interest to note that with thelower loss resistance ratio 0 to 14 ohms), theradiating efficiency is 7.14%, or 11.46 dB belowthe transmitter power delivered (excluding lineloss). With the 1- to 45-ohm resistance ratio (withthe higher loss coil and poor ground) the radiatingefficiency is 2.22%, or 16.53 dB below thetransmitter power. We thus have a 5.07 dB loss inefficiency in return for lowering the SWR from 3.5down to 1.1:1.

So, contrary to statements found innumerous articles which have been insisting thatwe believe otherwise, no significant improvementin efficiency can be obtained on the lowerfrequency bands by performing the matchingfunction between the feed line and the mobileantenna terminals when a low-loss loading coil isalready in use. The matching can be performedequally well at the input of the feed line, either bythe transmitter output tank itself, or by a separatematching network if the transmitter tank lackssufficient range, with the feed line connecteddirectly to the antenna terminals. (See refs. 4, 24,and 61.) Thus, as emphasized in the openingparagraphs of Part 5, the important point here,again, is that a flexible, open choice is available inour system design. Whether the matching which isrequired to transfer maximum power from thetransmitter into the line is to be performed at theinput end or the load end of the feed line is a choicewhich should be determined according to personalpreference of the operator. It should be based onhis convenience and the accessibility to adjustment,and not on an arbitrary, low SWR dictated for thewrong reason by a decree of a King who doesn'tunderstand his Subject! But wherever the matchingis performed during operation, the antenna that willradiate the strongest signal is the one with theloading coil that is capable of producing the highestSWR at resonance, with no matching at theantenna terminals, for the reasons explained above.

Finally, here is a suggestion which may behelpful in tuning a mobile antenna. The use of theGrid Dip Oscillator (GDO) for determining theresonant frequency of the coil and radiatorcombination can introduce errors of sizeablemagnitude, especially when disconnecting the lineto make the measurement, or when measuring atthe input terminals of the line. An SWR indicator

connected directly at the input of the feed line canprovide a more accurate indication of resonance,providing the instrument is reliable and accuratelycalibrated for the impedance of the fine (ref. 59),because minimum SWR for a given terminatingload of the type we're discussing occurs at theresonant frequency of the load, regardless of theline length (ref. Part 2 of this series, statement 24,and Part 5, the paragraph concerning "minimum-SWR resistance").

Low SWR for the Right Reasons

In concluding our remarks concerning lowSWR for the wrong reasons, it is of interest toknow that in TV broadcasting, where a long feedline is required to reach the antenna on a hightower, low SWR is an absolute necessity. However,the requirement here is primarily to avoid multiple,displaced images from appearing in the receivedpicture images which would result from reflectionson the feed line. Similarly, low SWR on the feedline is a necessity in fm stereo broadcasting toavoid cross contamination between the audiomodulation channels. However, in amateur radiowe do not have the problems encountered in TVand fm broadcasting.

To summarize the discussions of SWR andreflections as they pertain to amateur radiooperations, we have seen that we do not need a lowSWR on the antenna feedline:

a) to prevent reflected power fromdissipating in the transmitter, because nonedissipates in the properly coupled transmitteranyway, whatever the SWR;

b) to prevent feed-line radiation or TVI,because a mismatched load on the feed line doesn'tcause feed-line radiation or TVI;

c) to attain proper coupling to thetransmitter, because we can couple to or match theimpedance at the input terminals of the line,whatever the SWR. (A detailed enlightenment onthis crucial point has been promised for laterpresentation.)

And from reexamination of Figs. 9 and 10 itis evident that we do not need an SWR lower than2:1 on any line to avoid any significant loss inefficiency, or with SWRs considerably higher than2:1 when using feed lines having low attenuation.It would seem that there aren't too many reasonsfor needing a low SWR in amateur operations at hf

(ref. Part 1, p. 40 and Part 2, statements 11 through17). So let's see how we can briefly developrealistic SWR limits in relation to the attenuationvalues found in practical feed lines. Here are a fewtime-tested rules of thumb to use as guidelines.

1) When operation is near the dipole-resonant frequency, either 50- or 75-ohm feed linemay be used equally well. Depending on the heightabove ground, the antenna terminal resistance atresonance will fall somewhere between 50 and 80ohms, so the resulting mismatch with either lineimpedance is so small as to be inconsequential,despite arguments to the contrary from those whoare still afflicted with low-viswarmania. However,to obtain accurate readings with an SWR indicator,the impedance of the indicator must be compatiblewith the impedance of the line on which it's beingused.

2) A conjugate match placed anywhere onthe line between a mismatched load and thetransmitter compensates for the mismatch at theload, with the resulting effect that a conjugatematch now exists everywhere on the line (ref. 17, p.243). In other words, if a mismatched load ZL = R+ jX terminating the line is conjugately matchedsomewhere on the line, the reflection generated bythe complementary mismatch at the conjugatematching point causes the impedance looking intothe termination end of the line to change from ZC toZ = R - jX (see Part 4, pp. 22 and 23).

3) Now let's take advantage of the increasedbandwidth immediately available to us simply inour knowledge that nothing magical or miraculoushappens in "bringing the SWR down" to 1.0. Whenusing coaxial line to center feed a dipole, theoperation is usually for one amateur band only. Butnow we have the freedom to operate anywherewithin the entire band, letting the SWR climb towhatever value it should as the antenna terminalimpedance changes with frequency (but stillstaying within limits we will define presently).30 Tominimize the increase in mismatch and SWRresulting from the frequency excursion to either ofthe band ends, the dipole should be cut to resonatenear the center of the band. On the 75-80 meterband where the percentage of frequency excursionis the greatest, the mismatch at the band ends willbe somewhat less severe with a 75-ohm feed linethan with a 50-ohm line. Of the smaller sizedcables, RG-59/U is preferred here over RG-58/U,because the combination of the lower maximum

SWR and lower matched-line attenuation with theRG-59/U permits either a greater frequencyexcursion away from the self-resonant frequency ofthe dipole, or a longer line for the same loss factor.Of the larger size cable, either RG-8/U or RG-11/Ugives nearly equal results, because the matched-lineattenuation of the RG-11/U is a little greater thanwith RG-8/U, offsetting the gain resulting from itslower maximum SWR. However, the lowerattenuation of the larger cables permits either agreater frequency excursion or a longer line for thesame loss factor than the smaller cables,irrespective of their relative power-handlingcapabilities.

4) The smallest reduction in power that canjust barely be detected as a change in level at thereceiving station is 1.0 dB. So, to find the SWRwhich reduces the radiated power by 1.0 dB, firstuse Fig. 10 to find the attenuation per hundred feetof the correct feed-line type at the desired operatingfrequency. Then apply the correction for the actualfeed line. Now go to Fig. 9 and find the α-losscurve which corresponds to the value of the feed-line attenuation. Starting where that loss curvecrosses the SWR = 1.0 line, follow the curve to theright until 1.0 dB additional loss is indicated. Readthe SWR at this point. This is the SWR which willreduce the radiated power by the "just barelynoticeable" amount at the receiving station,compared with the signal that would be received ifthe line had been perfectly matched at the load.More exact data will be presented later, but thefollowing are fair rule-of-thumb values for dipoleSWRs to be expected at the band ends when thedipole is cut for resonance at the band center:

Freq. Max. SWR Value

3.5 to 4.0 MHz 5 or 6:1 (50-ohm line) 4 to 4.5:1 (75-ohm line)

7.0 to 7.3 2.5:114.0 to 14.35 2.0:121.0 to 21.45 2.0:128.0 to 30.0 3.0:1

Applying these data to Fig. 9 readily shows that itrequires feed lines of lengths substantially longerthan the average to lose enough additional powerfrom SWR ever to be noticed at the receivingstation. In other words, a full 1.0 dB of additionalloss will seldom be encountered, and therefore, no

"pile-up punch" will be sacrificed in obtaining theincreased operating-bandwidth flexibility.

5) At an SWR of around 4:1, the additionalloss due to SWR just equals the perfectly matchedline attenuation, thus, in effect, multiplying thematched loss by a factor of two. As an example,this statement translates: The power lost in 175 feetof RG-8/U, or in 87 feet of RG-59/U, at 4.0 MHzwith an SWR of 4:1 will have a "just barelynoticeable" difference compared to a line having noattenuation loss whatever! This is because theselines each have a matched-line attenuation of 0.5dB.

6) The SWR on the feed line may bemonitored to determine that the SWR is within thelimit based on the line attenuation, by placing theSWR indicator between the line-matching networkand the feed-line input terminals. But remember,the SWR remains on the line even after thematching network has been properly adjusted. Thematch between the transmitter and the matchingnetwork may be monitored with the SWR indicatorplaced between the transmitter and the network.The network is properly adjusted when the forwardpower is maximum and the power reflected fromthe network is zero. If the forward power readingsare the same as those obtained with a dummy load,and the reflected power reading is zero in bothcases, the input impedance of the network is thesame as the impedance of the dummy load. If theSWR indicator shows some power being reflectedfrom the matching network and the transmitter stillloads properly, obtaining further reduction of thereflected power is probably unimportant. Thisindication of reflected power is not showing an"SWR," but only the degree of impedancemismatch at the input of the network. If insufficientTVI rejection is obtained with the line-matchingnetwork alone, a conventional TVI filter may beused between the transmitter and the matchingnetwork with the same degree of effectiveness aswhen used in a line which is matched at the load.

Since it has now been shown that anyrequired matching can be performed at the input tothe feed line, instead of at the load, no SWRbandwidth limit for amateur use (such as thecommonly used, low, arbitrary 2:1 limit) isrealistic unless it is based on the attenuation of thespecified feed-line installation and the amount oftotal attenuation allowed. (The arbitrary 2:1 SWRlimit came into existence because the matching

range of most amateur transmitters has beenlimited to 2:1 by design, with economics beingconsidered more important than operationalflexibility. But simple line-matching networks asdescribed in the bibliography references can extendthe inherent matching range of the transmitters toaccommodate values of load impedance far beyondthe limits defined by a 2:1 SWR. If it were not forthe cost and space factors, these networks could bebuilt into the transmitters, giving us back thematching range we were accustomed to havingwith the old swinging-link method of coupling. Weweren't as conscious of SWR before the pi-netcoupling replaced the swinging link, because theconjugate matching at the line input then basicallyinvolved a simple adjustment of the link position toachieve the proper degree of coupling, and theretuning of the plate-tank capacitor to cancel thereflected reactance. Using this technique, we oftenloaded our transmitters into lines with high SWRwithout even knowing about the SWR. But withthe appearance of the SWR indicators after thedeparture of the link, we "discovered" SWR andthen learned how to misinterpret the meaning of theSWR readings.)

In conclusion, if the feed-line loss is withinyour acceptable limits at a given SWR level,determined from consulting Figs. 9 and 10, and thetransmitter can be adjusted to load and tuneproperly (either with or without an additional line-matching network), operate, and don't worry aboutthe SWR -- because you are now using realisticSWR for the right reasons!

Although reader response to this series ofarticles has been excellent, some have told theauthor, "Your story is interesting, but you'll neverconvince me that I won't get out better with aperfect 1.0 SWR." Now any reader who stillentertains any skepticism of these entireproceedings concerning SWR is reminded that theinformation presented herein is not simply arecitation of the ideas or opinions held by thewriter, but has been taken directly from theprofessional scientific and engineering literature(see extensive bibliography), and paraphrasedspecifically for the radio amateur with great carenot to change the meaning. Moreover, in strikingcontrast to the many differing opinions heard onthe subject during amateur discussions, there areno such differing opinions among the professionalsources, because among the professionals

(including text-book authors) the principlesinvolved are completely understood and are basedon true, proven scientific facts which are notsubject to divergent opinions as found in politics orreligion.

Apparently many have forgotten that thisstory was told for the amateur in QST no less thantwice prior to this series, by two well-knownexperts in this subject area. They are Mr. GeorgeGrammer, W1DF, engineer and retired former

Technical Editor of QST, and Dr. Yardley Beers,WØJF, formerly a professor of physics and Chiefof the Radio Standards Physics Division, NationalBureau of Standards, and presently SeniorScientist, Quantum Electronics Division of theNational Bureau of Standards. Their illuminatingcontributions, listed as references 6, 16, and 22 ofthe bibliography appearing at the end of Part 1 ofthis series, should be reviewed, even if it means atrip to the library the trip will be very rewarding.

Part 7: My Transmatch really does tune my antenna

The birth of this series touched off abombshell aimed at SWR misconceptions, emittingshock waves impacting on novices and old-timersalike. The myth believers among them were caughtwith their feed lines dangling at 1.05:1 -- and withthe low-viswarmaniacs screaming at them to "Getthat SWR down!" and Maxwell shouting "If ya do,it'll kill ya!" They didn't know whether to burntheir noise bridges behind them or to lock theirVFOs on the resonant frequency of the antenna.Probably the misconception causing the greatestshock of all concerns reflection loss -- the "lostreflected power" excuse offered by low-viswarmaniacs in defense of their insistence on lowvalues of SWR on the feed line. Thismisconception was blasted to extinction byexplaining the concept of conjugate matching in anunusual way -- from the viewpoint of reflectionsand wave action on the line. From this viewpoint(which involves both reflection loss and reflectiongain), the conjugate match concept emerged as abasis for understanding the lost-reflected-powermyth. However, while reflections from this seriesshow that the blast dispelled the myth for manyreaders, some are apparently jousting for anotherround. So let's take aim for the seventh round byreiterating a basic principle of matching, afterwhich we'll contemplate some enticing aspects ofline-matching networks. In this part we will see inan intriguing way how transmitters are matched tomismatched feed lines by the pi-section network(Fig. 12), or by an external network (Fig. 11).

The principle that maximum efficiency isobtained when a feed line is perfectly matched toan antenna -- no reflections on the feed tine, and a1:1 SWR -- is so well known it hardly needsrepeating. Even so, it is recited in practically everygood textbook on the subject. So it is important toappreciate that no statement in this series violatesthis principle, nor suggests any disagreement withit whatever. The misconceptions we are attemptingto clarify concerning, lost reflected power stemsimply from misuse of the perfect-match principlein its practical applications.

Ironically, textbooks may be a bitresponsible for this misuse. While extolling the

virtues of the perfect match, many authors fail toexplain how much (or how little) one loses whenthe load is mismatched, if compensated with theconjugate match. Those authors generally presentthe case for the ideal load match in terms of single-frequency operation, and practically ignore theunique, multifrequency operations of the radioamateur. We have frequency bands, not single,specific frequencies - and we want to operate ourVFOs anywhere within those bands. Since theantenna impedance changes as we changefrequency, relaxation of any antenna/feed-linematching restriction is necessary if we are to enjoysuch operating freedom. Although manyengineering texts discuss loss versus loadmismatch, few textbooks discuss multifrequencyantenna-matching situations where conjugatematching usually exists. Consequently, an overlyrigorous application of the perfect-load matchingprinciple has unwittingly been thrust upon us bydozens of misleading statements appearing invarious amateur journals, Add to this a prevalentmisconception of pi-network loading principles.Result? The "'lost reflected power" syndrome andthe mania for low SWR. Thus, providing guidancefor the amateur concerning match quality andefficiency in his multi-frequency operation is theprimary purpose of this series.

For example, the graph in Fig. 9, Part 6,displays transmission loss in dB versus SWR forvarious values of line attenuation. This graphshows that maximum, efficiency is indeed obtainedwith a perfect match. On the other hand, it alsoprovides dramatic evidence that when usingconjugately matched, low-loss feed line, thedifference between having either a perfect loadmatch or a moderately high SWR is practicallyinsignificant in terms of power transferred to theantenna. In other words, through the use ofconjugate matching, the antenna accepts themaximum available power from the transmitter,even when the feed line and antenna aremismatched, and with the antenna off resonance.31

The conjugate match allows us to tolerate this loadmismatch because the reflection loss caused by themismatch is compensated by reflection gain

provided by the conjugate match. Consequently,the transmitter is properly coupled to its desiredload impedance and the reflected power isconserved. Underlying these intrinsiccharacteristics of the conjugate match is systemresonance, which compensates for the effect of theoff-resonant state of the antenna.

For the myth believer who is stillunconvinced of the wondrous compensatingpowers of conjugate matching, let me quote Everitton the Maximum Power-Transfer Theorem fromclassical network theory (Ref. 17, page 49). Itsrelation to feed-line antenna matching is indicatedin the parentheses: "The maximum power will beabsorbed by one network (the antenna) fromanother (the feed line) joined to it at two terminals,when the impedance of the receiving (load)network (the antenna) is varied, if the impedancesof the two networks at the junction are conjugatesof each other." (Everitt then presents the proof.)

The expressions in Eqs. 12 and 13 (plottedin Fig. 9, Part 6) illustrate this theorem for the casewhere a feed line is the sending network. Theseexpressions show that when the networks areconjugately matched, meaning the transmitter istuned to the line, (1) there is no loss at theterminals joining the networks, (2) there is no lossin the sending network, the feed line, when thatnetwork attenuation is zero, and (3) if the sendingnetwork has attenuation other than zero, thetransmission loss results only from the attenuation.For further discussion the reader is referred to Part2, p. 21, Part 4, p. 25, mid Part 5, p. 163, of thisseries.

The wave action through which conjugatematching and reflection gain are achieved wasillustrated in Part 4 using the stub form of matchingto develop the wave action. The stub form wasused for the illustration because it is easy tovisualize. But since it's not a practical form to useif frequent changes in frequency are contemplated,let's see how conjugate matching is performedusing devices which are easily adjusted to performat any desired frequency. Such devices are theUltimate Transmatch (Ref. 41). a T network, L, pi,and other types of networks. We will see that thesedevices can perform the matching function at theinput of a feed line, and that the feed line can be ofany length, We will also see how, in someinstances, line-input matching is performed by thetransmitter tank circuit itself.

Match at the Line Input

Before going further the reader may ask,"'Why match at the input?" The answer is thatwithout matching at the feed-line input we havevery little operating flexibility. In the absence of aline-matching network we are restricted tooperating in a narrow portion of a band (especially80 meters) unless effective measures forbroadbanding the antenna itself have been taken.We are restricted because, as we deviate from theantenna-resonant frequency, the resulting increasein feed-line/antenna impedance mismatch istransferred to the line input as an increasedtransmitter/feed-line impedance mismatch. As aresult, the transmitter load impedance variesbeyond acceptable limits; the transmitter fails toload property, and it can be damaged byoverloading or by arc-overs caused byunderloading. These phenomena (plus anunawareness of the remarkable performancecapabilities of line matching) are largelyresponsible for the traditional low SWR mania. Onthe other hand, simple impedance matching at thefeed-line input provides stupendous improvementin operating flexibility because the matchingnetwork compensates for the impedance changes atthe feed-line input, and provides the correct loadimpedance for the transmitter at whateverfrequency we select within an entire band. Correctload impedance is obtained by simply adjusting thenetwork, conveniently located at the operatingposition.

So the next question is, "Why notbroadband the antenna and avoid having to retune amatching device when changing frequency?" Theanswer is that we can, but only to a limited degree.This is because, for example, broadbandingtechniques which would permit coupling theaverage amateur transmitter directly into the feedline over the entire 80-meter band (with noadjustments other than retuning the transmitter) arenot practical in the average amateur situation. Thisincludes the coaxial dipole (sometimes called a"double bazooka".) which, contrary to prevalentopinion, fails to deliver any significant bandwidthimprovement over a simple dipole when it is fedwith the usual 50-ohm feed line. While thisstatement may appear a bit incredible, a revealing

analysis by the writer appears in Ham Radio forAugust, 1976 (Ref. 62).

An explanation of how either a final tanknetwork or an external line-matching networkperforms conjugate matching from the viewpoint ofreflections is really a continuation of Part 4.However, to dramatize both the availability and theadvantages of matching at the feed-line input, wedigressed in Parts 5 and 6 to highlight some of thewrong reasons for using low SWR. These reasonsshow that, in the mania for obtaining low SWR (inthe feed line, we have often put misplacedemphasis on matching at the wrong end of the line.As stated previously, the principles of wavereflection found in stub matching are alsoapplicable to other matching schemes, such asseries quarter-wavelength transformers, L, T, pinetworks and so on. Since some of the conceptswe will be discussing were presented in detail inPart 4. you may wish to refer again to that section.

The Intermediate Role of theQuarter-Wavelength Transformer

How many remember the "Johnson QMatch"? And how many have used a quarter-wavelength section of 70-ohm line to match a 100-ohm resistive load to a 50-ohm feed line? Both areexamples of series λ/4 transformers. In addition torequiring an electrical length of 90-degrees,impedance matching is accomplished in thesetransformers because of a specific relationshipbetween their characteristic impedance ZC, and theimpedances being matched. To perform thematching, the transformer impedance ZC, must havea ratio relative to its input impedance Zi which isthe inverse of the ratio relative to its outputimpedance ZO, e.g.,

ZZ

ZZ

i

c

c

o

= (Eq. 14)

In other words, the impedance required of thetransformer is the geometric mean value of the twoimpedances being matched, stated simply by thewell-known expression

Z Z Zc i o= × (Eq. 15)

Both the impedance and the length of the λ/4transformer play important roles in clarifying the

principles underlying all forms of line-matchingnetworks. Since these roles are not generally wereunderstood, let's examine them.

It was shown in Part 4 that reflections playa necessary role in impedance-matching operations.We saw that conjugate matching is obtained bycanceling the reflections from a load mismatch bywave interference. The interference is set up bynew, separate reflections generated by a separatemismatch introduced at a desired matching point.The mismatch introduced at the matching point istailored to complement the load mismatch, so thatthe new reflection will have the same magnitudeand opposite phase (at the matching point) as thereflection generated by the load mismatch. Thereflection generated by this complementarymismatch is called either a complementary, or acanceling reflection. In stub matching, thecomplementary mismatch is introduced by the stub.While investigating complementary reflectionsgenerated by stubs, we observed the wave actionthrough which the λ/4 transformer performs thematching (Part 4, Table 1 and Fig. 7D). Whileanalyzing the effects of using various stub andtransformer lengths in matching a resistive load tovarious feed lines having different values ofimpedance, we found that when the feed-lineimpedance is equal to Zi (relative to transformerimpedance Z, and load impedance ZO, in Eqs. 14and 15), transformer length became 90 degrees, andthe stub length became zero. In this case thecomplementary reflection is generated by themismatch appearing at the junction of the feedlineand the transformer. This mismatch results fromthe abrupt change in impedance (Zi to ZC)encountered by the forward wave as it propagatesout of the feed line (Zi) into the transformer (ZC).This mismatch is complementary to the resistiveload mismatch in magnitude because the ratiobetween the feed line and transformer impedances(Zi / ZC,) is the same as between the transformerand load impedances (ZC / ZO). Thus the reflectionsgenerated by both load mismatch and transformer-input mismatch are equal in magnitude (as requiredto obtain perfect cancellation), because bothmismatches are equal in magnitude.

Since the input and output mismatches arephysically separated by 90 degrees, they are alsocomplementary in relation to the phase of thereflections appearing at the matching point (alsorequired for cancellation). The wave reflected by

the input mismatch has zero distance to travelrelative to the matching point. But the 90-degreeseparation results in a travel of 180 degrees for thewave reflected by the load mismatch -- 90 degreesfrom the input to the load, plus the 90-degree returntrip. Thus the load-reflected wave arrives at thematching point with a 180-degree phase differencerelative to the input-reflected wave. We now havetwo complementary reflected waves -- equal inmagnitude and opposite in phase at the matchingpoint. Consequently the two waves mutuallycancel, resulting in total reflection of both wavesinto the transformer to propagate in the forwarddirection, as explained in Part 4. The voltage andcurrent components of both re-reflected waves arein phase with their corresponding components ofthe source wave; hence a conjugate match appearsat the transformer input, all of the power reflectedfrom the load mismatch which reaches thetransformer input is again on its way to the load,and no reflected wave appears on the feed line,

This λ/4-transformer matching actiondeserves serious study. Why? Because it providesan intermediate step in understanding howmatching can be achieved at the input of a linetransformer of any random length, and which mayhave any impedance for its terminating load, suchas the complex impedance Za = Ra + jXa of amismatched, off-resonant antenna. And this linetransformer is none other than the feed line somany strive to operate with no reflections bysimply restricting its load to a matched, resonantantenna!

Input Line-Matching Networks

Now let's examine external line-matchingnetworks (such as the Ultimate Transmatch) fromthe viewpoint of conjugate matching withreflections. Referring to Fig. 7D of Part 4, wereplace the 90-degree transformer (T) with thecombination of an adjustable network and a line ofrandom length to connect the mismatched load(antenna) to the network. This arrangement isshown in Fig. 11. The line connecting the networkto the antenna will now be called the "transformerline." The line connecting the transmitter to thenetwork. is the "feed line,"' as before. Thematching point is defined as the junction of thefeed line and the input of the network.

In a manner which will be explained later,the transformer line (which can have any value Z'c)transforms the complex antenna impedance Za = Ra

+ jXa to a second impedance Z2 = R2 + jX2 at thetransformer-line input. The network thentransforms Z2 to a third impedance Z3 = R3 + jX3 atthe matching point, When the network is correctlyadjusted, Z3 is a pure resistance equal to the feed-line impedance ZC. (Z3 = R3 + j0 = ZC) Thus theantenna impedance Za is conjugately matched tothe feedline impedance ZC, which is a proper loadfor the transmitter (Ref. 17, p. 243). Without thenetwork the transmitter load would be impedanceZ2, well known for deviating far beyond the rangeof matching capability for most transmitters.However, by adding the network we obtain theconjugate match by simply adjusting the networkto transform R2 to equal the feed-line impedance ZC

at the matching point, and to cancel anytransformer-line reactance X2 to zero at thematching point.

Let's now examine the transformation ofimpedance Z2, more closely. The variations ofresistance R2 and reactance X2 of impedance Z2 aredependent on three different factors: Antennaimpedance Za and both the length and theimpedance Z'c, of the transformer line. A study ofPart 4, pages 27-28, will show that for givenantenna and transformer-line impedances, a lengthof line can be found that will make R2 = ZC, butwill yield reactance X2 which requires canceling toobtain a match (for example, with a stub as shownin Fig. 7A or 7B). Another length can be found thatwill make X2 become zero, but now R2, will notequal ZC (still no match). There is no length thatwill yield Z2 = R2 + j0 = ZC. This situationillustrates the typical, endless cat-and-mouse gamewe play in trying to load the transmitter bychanging line lengths. A fanciful solution to thisproblem would be a line of variable length, plus adevice for dumping the unwanted reactance. Sohow can we possibly solve the problem using afixed, random-length line? Because, as we nowdiscover, the line-matching network is a starperformer! We know from elementarytransmission-line theory that a transmission line ismade up of an infinite number of tiny distributedseries inductances and shunt capacitances, so itshould not be surprising that we can adjust theelectrical length of a line of a given physical lengthby adding-lumped inductances or capacitances.

Indeed, by a proper selection and adjustment ofreactances arranged in an appropriate L, T, or piconfiguration we can simulate a line having anydesired electrical length without specifying anyphysical length whatever (Refs. 8-13; .19, p. 115:21; 22; 24; 30; 31; 41; 61; 63).

When a matching network is adjusted toobtain a conjugate match between the antenna andfeed-line impedances, it performs the followingtwo feats. First, it creates the effect of stretchingthe electrical length of the transformer line to makeit reach the matching point, so that the resistance R2

(at the input of the physical transformer line) istransformed to R3 = ZC. And second, it introducesreactance -X3 to cancel the stretched-transformer-line reactance X3 appearing at the matching point,in the same manner as a stub would perform in stubmatching.

The introduction of reactance - X3 at thematching point constitutes a complementarymismatch which generates a canceling reflectionhaving equal magnitude and opposite phase relativeto the reflected wave arriving at the matching pointfrom the mismatched antenna. The matchingnetwork has thus provided the proper overalltransformer length to obtain both the desiredtransformation of the resistance component, and acanceling phase relation between the load-reflectedwave and the canceling reflected wave. And inaddition, the network has provided thecomplementary mismatch (-X3.) which generatedthe canceling wave. Consequently, the network hasalso transformed a dream into reality by conjuringup the fanciful variable-length line and means fordumping the unwanted reactance,

An ideal setup for observing the actionwhile performing the tuning adjustments of thenetwork includes an rf ammeter in the transformerline to indicate line current (a meter in eachconductor if using balanced, two-wire line), and adual-meter reflectometer to indicate forward andreflected power simultaneously in the feed line. Itis important that the reflectometer be adjustedinitially to indicate zero reflected power with thefeed line terminated in a pure resistance equal tothe feedline impedance. The tuning adjustments arecomplete when we obtain maximum current in thetransformer line simultaneously with maximumforward and zero reflected power in the feed fine.Of course, because of standing waves on thetransformer line, we will obtain different values of

line current depending on where along the line theammeter is inserted. However, our only interest isin seeing changes in relative current to indicatewhen maximum network output occurs during thetuning adjustments; thus neither absolute linecurrent, nor where the meter is inserted, isImportant.

The simultaneous indication of maximumcurrent in the transformer line and zero reflectedpower in the feed line denotes four significantfactors for comparing the wave actions involved inconjugate matching with a line-matching networkand matching with the λ/4 transformer describedearlier. First, proper network adjustmentestablishes the complementary mismatch betweenthe feed-fine termination and the network inputwhich produces the canceling reflection at thematching point. Second, at the matching point thecanceling reflection is equal in magnitude andopposite in phase relative to the load-mismatchreflection and, the canceling reflection and theload-mismatch reflection nullify each other at thematching point. Consequently, a purely resistiveimpedance equal to the feed-line impedance ZC

appears at the network-input terminals, whilereflections and a standing wave remain on thetransformer line. And fourth, observing thetransformer-line current rising to maximum whilethe reflected power in the feed line drops to zeroprovides visible evidence that the power reflectedfrom the load mismatch is indeed rereflected by thecomplementary mismatch at the matching point.

Incidentally, it is good practice to have anammeter permanently connected in the transformerline. Here's why. If your matching networkeffectively comprises more than one L-section, youcan generally obtain a match (zero reflected powerin the feed line) with several differentcombinations of network L and C tuning. However,minimum network loss (which corresponds tohighest maximum transformer-line current) usuallyoccurs while using the maximum C and minimumL at which a match can be obtained. Monitoringtransformer-line current during tune-up lets youselect the L-C combination yielding the highestoutput line current; it also ensures againstinadvertently obtaining an L-C combination whichgives a false indication of a match, but yields littleoutput and instead heats the network inductance.To ensure quick resettability and to minimize on-the-air tune-up tune, L and C settings for the best

combination Should be logged whenever thenetwork is tuned to a new frequency. Thetransmitter should be tuned initially into a dummyload, then the network tuned into the antenna usingthe lowest power at which the reflectometerprovides a satisfactory indication.

Tank Circuit Line Matching

Let's now examine the case where the finalamplifier tank circuit performs the line-matchingfunction. To differentiate from the external networkwe will call this network the "tank network."Comparing Figs. 11 and 12, we see that, in general,the transformations of impedance from Za to Z3 areidentical whether using the tank network or theexternal network. The principal difference is therange of the transformations performed by the twonetworks. With the external network, impedance Z2

is transformed to a value of Z3, matching the feed-line impedance ZC. When the tank network is usedalone, impedance Z2 is transformed to matchdirectly the load impedance ZL of the generator.Now a requirement for a generator (tube ortransistor) to deliver its maximum available powerinto a load (the loaded tank circuit) is for it to see

an impedance which we call, the optimum loadimpedance, ZL, (not the same as internalimpedance, Zg.). In practice, impedance ZL, isusually resistive, so that ZL = RL + j0. Thus thegenerator is property loaded when it seesimpedance Z3 = RL + j0 (which is R3 + j0) lookinginto the tank network. The generator is underloadedwith R3 larger than RL, overloaded with R3 less thanRL and it will have higher than normal dissipationif Z3 contains any reactance X3.

When using the tank network alone toperform the matching, the impedance value Z3 isdetermined by two factors: The impedance value ofZ2 loading the network, and the impedancetransformation ratio of the network. Thetransformation ratio is somewhat variable (usingthe tuning and loading capacitors, C1 and C2), andproviding a range of impedance-matchingcapability. This matching range allows impedanceZ2 to be any value which the network can transformto the impedance Z3 = RL + j0 by adjusting thetuning and loading controls.

Now we can summarize the principaloperating conditions involving transformation ofantenna impedances to the optimum loadimpedance for the generator.

Fig. 11 - Conjugate matching with a Transmatch or other matching network external to the transmitter. The impedance of the feedline, ZC, is usually 50 or 75 ohms in most amateur applications, but the impedance of the transformer line, Z'C, may be any value,and may be of open-wire or coaxial type. The matching network as A is a modified Ultimate Transmatch (Ref. 41), as shown at B.NOTE: In the redrawn circuit at B (center) the shunt input capacitor is not needed to obtain a match, and the capacitor range may beincreased by reconnecting as shown at the right.

Case 1Antenna impedance Za = Ra + j0 (resonant) ismatched to the transformer line, and the tanknetwork is used alone -- no external matcher: Hereantenna impedance Za, equals the impedance of thetransformer-line, ZC, the SWR on the line is 1:1,and impedance Z2 is equal to the line impedance. Ifthe transformer-line impedance ZC is the commonlyused 50 ohms, the tank network yields the propergenerator-loading RL + j0 with the same tankadjustment settings as obtained when tuning upwith a 50-ohm dummy loadCase 2

Antenna is operated somewhat offresonance. Its impedance Za = Ra + jXa yields amismatch to the transformer line so as to transformZa to an impedance Z2 that is within the matchingrange of the tank network (for transformation to Z3

= RL + j0). Again the tank network is used solo.

Case 3Antenna operated off resonance beyond

matching range of tank network. Here the tanknetwork is used in conjunction with an externalnetwork. An impedance Z3 = RL + j0 cannot beobtained with the tank network alone, and thegenerator would be either underloaded, overloaded,or reactively loaded. We remedy this situation byinserting an external line-matching network (as inFig. 11), which transforms impedance Z2 to animpedance Z3 that is within the range the tanknetwork can transform to RL + j0.Case 1 needs no comment, so let's refer to Fig. 12and examine the action in the tank network as itperforms the matching function under theconditions of Case 2. As stated earlier, when ourgenerator is the commonly used Class AB, B or C

amplifier, its optimum load impedance ZL and itsinternal impedance Zg are not the same. Althoughthe explanation is beyond the scope of this article,it can be shown that Zg effects only a light, high-impedance loading on the input of the tanknetwork. Thus the input of the network (thematching point) is practically open circuited toenergy traveling toward the generator.Consequently, waves reflected from a mismatchedantenna (which cause impedance Z2 to deviate fromZC ) enter the network at its output terminals andbecome almost totally reflected on arrival at theopen-circuited input. When the network is tuned toresonance, voltage and current components of thereflected waves are reflected in phase with thecorresponding source-wave components emanatingfrom the generator. Thus the generator sees aresistive load Z3 = R3 + j0, and the reflected poweradds to the generator power. This is why adirectional wattmeter (or reflectometer) indicates ahigher forward power than the generator is actuallydelivering when reflections are present on thetransformer line. (See Part 4.)

Adjustment of capacitance C2 (the loading-control capacitor) to the value which providesoptimum generator loading adjusts the network totransform R2 to R3 = RL. When R2 changes,following a change in antenna impedance Za, C2can then be varied to modify the networktransformation ratio to transform the new value ofR2 to R3 = RL. The tuning capacitor, C1, is thenreadjusted to return the network to resonance an weagain have optimum generator loading. The rangeof R2 values which can be transformed to R3 = RL

by varying C2 is determined by the designparameters of the network (Refs. 4, 63, 64).

Fig. 12 - Conjugate matching with the transmitter final amplifier tank circuit. ZL is defined as the optimum load impedance of thegenerator - the impedance into which the generator delivers maximum available power. This impedance must not be confused withthe internal impedance of the generator, Zg; they are not the same.

However, when the line input impedance Z2

loading the network contains reactance X2, thisreactance shifts the available range of capacitanceprovided by the loading control capacitor, C2. Thisshift occurs because reactance X2 appears inparallel with the loading capacitor reactance.Consequently, for a proper setting of the loadingcapacitor where X2 = 0, a capacitive X2 increasesthe effective capacitance C2. (decreasing loading),while an inductive X2 decreases C2 (and increasesloading). Thus, to obtain a proper loading when X2

is present, the setting of the loading-controlcapacitor must be shifted from the X2 = 0 positionto compensate for the additional line-reactance XL.Providing R2 is within the matching range, properloading will be attainable as long as X2 doesn'texceed the value which the loading-controlcapacitor range can accommodate to maintain thevalue of C2 required to transform R2 to R3 = RL.

If reactance X2 is too large for the loadingcapacitor to compensate, we have conditions asdescribed in Case 3. However, there are simpleremedies for this condition if R2 is within matchingrange of the tank network, but X2 is not. Here wecan simply add a compensating reactance in serieswith the rf output (between points A and B in Fig.12) which cancels the line reactance X2. On theother hand, if the resistance component R2P of theequivalent parallel-circuit of impedance Z2 iswithin the matching range, the compensatingreactance should be added in shunt across the rfoutput (between points A and C). Whether thecompensating reactance should be capacitive orinductive can be determined experimentally bytrying first one kind or the other to see whetherloading is improved or worsened. The correct kindand value of compensating reactance has beenfound when proper loading can be obtained with aconvenient setting of the loading-control capacitor.An excellent discussion of this subject by Grammerappears in the literature (Ref. 4, Part 3). Whenusing this matching technique with the tanknetwork alone, an adjustment of transformer linelength can be of great assistance in obtainingvalues of impedance Z2 which are most favorable tothe matching range of the network. This subject ofimpedance transformation versus line length willbe discussed in a later installment in this series ofarticles.

When operating under the conditions ofCase 2, tuning up into a dummy load requires

special care; it can be troublesome since the actualload often differs widely from the dummy-loadvalue. If the tank network is first tuned into thedummy load and then switched to the transformer-line input impedance Z2 (Fig. 12), the tank networkmust be retuned to the new impedance Z2! Failureto retune to the actual operating impedance Z2 aftertuning up into the dummy load results in animproper load impedance Z3 for the generator -- itis no longer Z3 = RL + j0 because the tank-networkloading has changed. Without retuning, thegenerator is then either underloaded, overloaded orreactively loaded. In addition to the possibility ofdamaging the amplifier if operated in this mistunedcondition, you also lose power! Because of thismistuning, the generator delivers less power by theamount of the reflection loss resulting from the linemismatch, as shown in Part 6, Fig. 9, on the curvelabeled "Reflection Loss Without ConjugateMatch" at the appropriate SWR ordinate. Forexample, if the line SWR were 3:1, the generatoroutput drops by 25 percent while it sees theimproper load. On the other hand, retuning thenetwork to the actual operating impedance Z2

establishes a conjugate match that restores theproper generator load impedance RL + j0, and thegenerator again delivers its maximum availablepower. (See Part 6, pp. 14-15.)

But you ask, “How do we determine whenproper loading is obtained, or that impedance Z2 iswithin the matching range of the network?" Theanswer is simply by completing a normal tuningand loading operation in which you can obtain thesame plate-current, plate-current dip (and screen-current) reading as those obtained with the dummyload. However, tuning- and loading-controlsettings, and the relative power (output voltage)will generally differ from those obtained with thedummy load, depending on how much Z2 differsfrom 50 ohms. If normal plate current (and dip)cannot be obtained with any setting of the loadingcapacitor, Z2 is outside the matching range and wehave conditions as defined in Case 3. If properloading cannot be obtained using the simple seriesor shunt reactance compensating techniquedescribed earlier, then a more complex network isrequired. However, I would like to emphasize thatany value of Z2 = R2 + jX2 can be transformed to asuitable value for loading the tank network byselecting a proper network configuration (Refs. 19.p. 115; 30; 61)

The range of impedances Z2 which the tanknetwork will transform to the value equal to thegenerator impedance ZL = RL + j0 raises aninteresting point concerning the tuning procedurefor the external network. The usual practice is totune the tank network with the dummy load, andthen switch in the external network and antennaand tune for zero reflected power in the feed line(not the transformer line). This procedure should befollowed if there is a filter in the feed line.However, in the absence of a filter, it is necessaryto adjust the external network only for an input

impedance Z3 which will bring it within thematching range of the tank network. This can be atime-saving feature when changing frequenciesduring contest operation! If both tank and externalnetworks are adjusted for optimum match atmidrange of the intended frequency excursions,only the tank network needs retuning with changesin frequency, providing the frequency excursionsare within the range in which the external networkyields a load impedance that the tank network cantransform to RL + j0.

Bibliography

1) The Radio Amateur's Handbook, any recentedition.

2) The ARRL Antenna Book, I 0th edition, pp.70-80 and 90-136. [EDITOR'S NOTE: Pagenumbers in the more recent 11th and 12th editionswill not be significantly different for discussions ofpertinent subjects.]

3) Grammer, "The Whys of TransmissionLines," QST, in three parts; Part 1, January 1965,p. 25; Part 11, February 1965, p. 24; Part 111,March 1965, p. 19.

4) Grammer, "Simplified Design of Impedance-Matching Networks," QST, in three parts; Part 1,March 1957, p. 38; Part 11, April 1957, p. 32; Part111, May 1957, p. 29.

5) Grammer, "Antennas and Feeders," QST, inthree parts; Part 1, October 1963, p. 30; Part 2,November 1963, p. 36; Part 111, December 1963,p. 53. [EDITOR'S NOTE: Any publishedreferences to a Part 4 of this series are erroneous.]

6) "Standing Waves - Good or Bad?",Technical Topics, QST, December 1946, p. 56.

7) Goodman, "My Feedline Tunes MyAntenna! ", QST, March 1956, p. 49.

8) McCoy, "Choosing a Transmission Line,QST, in two parts; Part 1, December 1959, p. 42;Part 2, February 1960, p. 40.

9) McCoy, "Antennas and Transmatches,"QST, October 1964, p. 18.

10) McCoy, "A Versatile Transmatch," QST,July 1965, p. 58.

11) McCoy, "A Completely FlexibleTransmatch for One Watt to 1000," QST, June1964, p. 39.

12) McCoy, "A Transmatch for Balanced andUnbalanced Lines," QST, October 1966, p. 38.

13) McCoy, "When is A Feed Line Not A FeedLine?", QST, August 1965, p. 37.

14) Drumeller, "The Mismatched rfTransmission Line," 73, November 1969, p. 68.

15) Smith, "The 'Mega-Rule,' " QST, March1969, p. 24.

16) Beers, "Match, or Not To Match?", QSTSeptember 19 5 8, p. 13.

17) Everitt, Communication Engineering, 2ndEdition, McGraw-Hill, New York, 1937.

18) Johnson, Transmission Lines andNetworks, McGraw-Hill, New York, 1950.

19) Smith, Electronic Applications of the SmithChart in Waveguide, Circuit and ComponentAnalysis, McGraw-Hill, New York.

20) Brown, Lewis, and Epstein, "GroundSystems as a Factor in Antenna Efficiency,"Proceedings of the IRE, June 1937, p. 753. 1NOTE: Discusses ground resistance vs. number ofradials - a classic work on which FCC standards fora-m broadcast ground systems are based.)

21) Smith, "Getting the Most Into YourAntenna," QST, July 1952, p. 21.

22) "Impedance Matching With an AntennaTuner," Technical Topics, QST, October 1946, p.38.

23) Ferber, "Coaxial Cable Attenuation," QST,April 1959, p. 20.

24) Leo, "An Impedance- Matching Method,"QST, December 1968, p. 24.

25) Hall, "Smith-Chart Calculations for theRadio Amateur," QST, in two parts; Part 1, January1966, p. 22; Part 11, February 1966, p. 30.

26) Cholewski, "Some Amateur Applicationsof the Smith Chart," QST, January 1960, p. 28.

27) Amis, "Antenna Impedance Matching,"CQ, in two parts; Part 1, November 1963, p. 63;Part 11, December 1963, p. 33.

28) Fisk, "How To Use the Smith Chart," HamRadio, November 1970, p. 16. (Also see ref. 29.)

29) Maxwell, "Correspondence on How to UseThe Smith Chart," Ham Radio, December 197 1, p.76 (for corrections to ref. 28).

30) Smith, "L-Type Impedance TransformingCircuits," Electronics, March 1942, p. 68.

31 ) Gordon, "L Networks for Reactive Loads,"QST, September 1966, p. 30.

32) Schultz, "Broad-Band Balun," CQ, August1968, p. 66.

33) Reference Data for Radio Engineers, 4thEdition, Federal Telephone and Radio Co., p. 564.

34) "Antenna and Transmission Line Quiz,"QST; Questions, July 1965, p. 19; Answers,August 1965, p. 55.

35) Slater, Microwave Transmission, McGraw-Hill, New York, pp. 7-69.36) Orr, "Broad Band Balun For A Buck," CQ,February 1966, pp. 42, 45.

37) King, Transmission Line Theory, McGraw-Hill and Dover Publications, Inc., pp. 77-83.

38) Bruene, "An Inside Picture of DirectionalWattmeters," QST, April 1959, p. 24.

39) Radio Research Laboratory Staff, HarvardUniversity, Very High-Frequency Techniques, Vol.1, Edition 1, McGraw-Hill, pp. 10-15.

40) Hall, "Accuracy of S.W.R. Measurements,"QST, November 1964, p. 50.

41) McCoy, "The Ultimate Transmatch,"QST, July 1970, p. 24.

42) Mumford, "Directional Couplers,"Proceedings of the IRE. February, 1947, p. 160,

43) Skilling, Fundamentals of Electric Waves,2nd Edition, p. 123; John Wiley and Sons, NowYork, 1948.

44) Shallon, "The Monimatch and S.W.R.,"QST for August, 1964, p. 54).

45) McCoy, "Is a Balun Required?" QST,December, 1968, p 29.

46) Hall and Kaufmann, "The Macromatcher,An rf Impedance Bridge for Coax Lines," QST,January, 1972, p. 14.

47) Brown and Woodward, "ExperimentallyDetermined Impedance Characteristics ofCylindrical Antennas," Proc. of the IRE, April,1945, p. 257.

48) King, Theory, of Linear Antennas, pp. 169-176, Harvard Univ. Press, Cambridge, Mass.

49) King and Blake, "The Self-Impedance ofthe Symmetrical Antenna," Proc. of the IRE, July,1942, p. 335.

50) Sevick, "The Ground-Image VerticalAntenna," QST, July, 1971, p. 16.

51) Sevick, "The W2FMI Ground-MountedShort Vertical," QST, March, 1973, p. 13.

52) Kraus, Antennas, McGraw-Hill, New York.53) Jordan, Electromagnetic Waves and

Radiating Systems, p. 415, Prentice-Hall, NewYork. 54) Fayman, "A Simple Computing SWRMeter," QST, July, 1973, p. 23.

55) Anderson, "SWR's Significance," CQ,October, 1970, p. 8.

56) Smith and Johnson, "Performance of ShortAntennas," Proc. of IRE, October, 1947, p. 1026.

57) Laport, Radio Antenna Engineering,McGraw-Hill, New York.

58) Jasik, Antenna Engineering Handbook,McGraw-Hill, New York.

59) Schreuer, "Notes On Directional SWRIndicators," Ham Radio, December, 1969, p. 65.

60) Belrose, "Short Antennas for MobileOperation," QST, September, 1953, p. 30.(Information also contained in The Mobile Manualfor Radio Amateurs, ARRL, now out of print.)

61) Leo, "How To Design L-Networks, "Ham Radio, February, 1974, p. 26.

62) Maxwell, "A Revealing Analysis of theCoaxial Dipole Antenna," Ham Radio, August,1976, p. 46.

63) Gray and Graham, Radio Transmitters.McGraw-Hill Book Company, 1961, pp. 115-133

64) Schottland, "Pi Networks as CoupledCircuits," Electronics, August, 1944.

Footnotes

2.Forward, or incident power equals line-inputpower plus reflected power. Line-input powerand incident power are equal only when the lineis perfectly matched at the load, for zero reflectedpower.

3 Transmitter manufacturers could easily havehelped put an end to this myth by including twoor three short paragraphs in their instructionmanuals explaining the maximum and minimumcoupling limits of the pi network, and how toextend them with an external capacitor, instead ofthe terse, uninformative "WARNING - DO NOTEXCEED 2:1 VSWR," and so forth.

4 Many baluns have high leakage reactance, andthus cannot provide a true one-for-one impedancetransfer. This reactance inserted between theantenna and its feed line can either improve orworsen the match, depending on the magnitudesand signs of the leakage and antenna-terminalreactances.

5 To satisfy the conjugate conditions when thegenerator is a Class B or C amplifier, theimpedance replacing the generator must be madeequal to its optimum load impedance, the loadinto which the generator delivers its maximumpower to the loaded tank. For a Class C amplifierthis is roughly twice its internal impedance. Thereason for this difference may be more fullyappreciated when we consider that the classicalnetwork generator has a maximum efficiencylimit of 50 percent because it delivers itsmaximum available power when its loadimpedance equals its internal resistance. But inthe Class C amplifier the effective internal acimpedance is about half of its optimum loadimpedance because the current pulses whichexcite the loaded tank are of high peak value andshort duration, while the instantaneous anodevoltage during current flow is very low. Thisresults in proportionately less power lost in thegenerator and more delivered to the load,enabling it to operate at an efficiency as high as75 percent or more.

6 With a conjugate match all reactance has beencanceled, leaving the reflected voltage polarityeither pure in- phase aiding, or out-of-phasebucking the source voltage. If bucking, theexplanation of the text is sufficient, because asource of smaller voltage can never cause areverse power flow through a larger voltagesource. Whether bucking or aiding, if the loadingadjustment leaves the generator impedance lowerthan the line-input impedance, the reflected waveis still totally rereflected, but the generator isundercoupled and not delivering all availablepower. But any loading adjustment or matchingerror which leaves the fine-input impedancelower than the generator impedance results inovercoupling, or overloading, and this, notreflected power, directly causes excessivegenerator dissipation and lowered efficiency.

7 Detailed instructions (plus examples) oncalculating the additional loss because of SWRfor given line attenuation and SWR values willbe presented later. The data may also be takenfrom part 1, Fig. 1, which comes from theTransmission Lines chapters of The ARRLAntenna Book and the ARRL Handbook, or frombibliography ref. 33 p. 573.

8 Smith charts may be obtained at most universitybook stores. They may be ordered (50 for $2.50postpaid when remittance is enclosed) fromPhillip H. Smith, Analog Instruments Co., P.O.Box 808, New Providence, NJ 07974. For 8 1/2 xclinch paper charts with normalized coordinates,request Form 82-BSPR. A brochure of Smithcharts and accessories will be sent upon request.NOTE: Smith charts with 50-ohm coordinates(Form 5301- 7569) are available at the sameprice from General Radio Co., West Concord,MA 01781.

9 Footnote 9 was the Editor's note, included withinthe text, for continuity

10 Reference ASA Y10.9, 1953 (AmericanStandards Association). Prior to the adoption ofthis standard, either Γ or k was frequently used toindicate the reflection coefficient, while ρ was

often used for standing-wave ratio. In studyingthe literature, the reader should exercise care toavoid confusing symbols used before and afteradoption of the standard.

11 Remember that motor-generator action resultsfrom mutual motion between a field and aconductor. Here the field is changing eventhough the conductor is not moving, as in atransformer.

12 The characteristic impedance, ZC of lossless linehas zero reactance, and low-loss line has so littlereactance that it is neglected.

13 Incident voltage and current should not beconfused with line voltage and current, becausethey are not the same except when the line isperfectly matched and no reflections exist.

14 Smith charts my be obtained at most universitybook stores. They may be ordered (50 for $2.50,postpaid when remittance is enclosed) fromPhillip H. Smith, Analog Instruments Co., P.0.Box 808, New Providence, NJ 07974. For 8-1/2X 11-inch paper charts with nomalizedcoordinates, request Form 82-BSPR. A brochureof Smith charts and accessories will be sent uponrequest. Smith charts with 50-ohm coordinates(Form 3801-7569) are available at the same pricefrom General Radio Co., West Concord, MA01781. Readers who are unfamiliar with theSmith chart are encouraged to consult thebibliography of Part 1 -for references; they willfind the chart a valuable tool (See ref. 19, 25. 26,27, 28, 29)

15 The relative phase angle between the incident andreflected voltage waves is the angle of thereflection coefficient, referenced on the incident.See the "Angle of Reflection Coefficient" scalearound the perimeter of the impedance plottingportion of the Smith chart.

16 On the chart the dashed lines show the vectorsafter 180 degrees of travel on the line.

17 Woods, "Power in Reflected Waves," HamRadio, October, 1971, and Woods'correspondence on same, Ham Radio, December,1972, p, 76.

18 DeLaMatyr, "Reflections on Reflected Power,'"Technical Correspondence, QST for November,1972, P. 46.

19 Line Voltage, E, and line current, I, are theresultants of the forward and reflected E (+ and -)and I (+ and -) components (Fig. 4). Line E and Irespectively, are measurable with a simple rfvoltmeter across the line and an ammeter inseries with the Line. Both quantities may be seento vary along the line with the reflections (Fig. 6),and are generally reactive except for the twospecific cases where they are in phase.

20 See point 3, Part 1 of this series (p. 38 of QST forApril, 1973).

21 Alternatively it may be a short-circuit, dependingon line and load conditions, which will beexplained later.

22 We are rather accustomed to thinking of 50 ohmsas a standard system impedance because of the*preponderance of rf components and coaxial linesfor that impedance. However, many calculationsare greatly simplified by using normalizedimpedances, in which all impedance values havebeen divided by the system impedance. Thesystem impedance is usually taken as thecharacteristic impedance of the transmission line,ZC. Normalizing an impedance by dividing it byZC amounts to a change of scale such that the unitof impedance is ZC ohms, rather than 1 ohm. TheSmith chart Vector Graph, Fig. 4, uses thenormalized system to take advantage of thesimplification in the calculations. To obtainnormalized values occurring in a system of anyimpedance, simply divide all impedances by thesystem impedance. For example, in a 50-ohmsystem 150 ohms becomes 3.0 ohms. To convertback to the 50-ohm system, simply reverse theprocess and multiply the normalized values by50. For example, the normalized impedance Z =0.6 - j0.8, found at L = 45°, becomes Z = 30 -j40. In the drawing for Fig. 6A, p. 24, the� resultant vector should be shown with a value

of ρ = 0.5 ∠ 0° rather than 60°. In the captionfor Fig. 7, p. 27, the last few words should read".. . stub length goes to zero.

24 See footnote 23.

25 Glanzer, "More Words on Antennas," CQ July,1957, p. 40; see p. 48 for material beingreferenced.

26 See footnote 25.

27 Houghton, "Convert SWR Into Watts," CQ, June,1970, p. 36. Here it is stated that reflected poweris lost, accompanied by a nomograph forconverting SWR into percent reflected power "tomake it easier to determine just how much poweris lost." The reader is invited to read an excellentrebuttal to this nomograph presentation byAnderson, VE3AAZ (ref. 55), also in CQ,October, 1970, p. 8.

28 Scherer, "CQ Reviews: The Knight Kit P-2 SWRPower Meter," CQ, March, 1963, p. 31.Reference is being made to data on p. 90 of thatissue, where 100% minus the reflected power iserroneously shown as "useful" power. Thesucceeding paragraph on p. 90 also statesincorrectly that the SWR indicator must beplaced at a multiple of a half wavelength from theload to indicate the true SWR. The exampledemonstrates that the SWR indicator was eithernot properly adjusted to the impedance of the

line, or that it was unreliable. See Part 2 of thisseries, statements 21 and 23, and refs. 38, pp. 25 -26; 59.

29 Many amateurs unknowingly select loading coilsof low Q because "they produce a lower SWRthan any other types of coils." They do, indeed,produce a lower SWR, because of their higherloss resistance, explained in the text.

30 An exception is that satisfactory operation maybe obtained on 15 meters with a 40-meter dipole.

31 Remember, impedance mismatch does not causecurrent to flow on the outside of the coaxial feedline. (See Part 5, QST, April, 1974, p. 27.)

Feedback – Part 4QST November 1973, pg. 46

Was it gremlins which crept into Part IV ofMaxwell's series of articles, "Another Look atReflections" (QST for October, 1973)? A portionof the information in footnote 22 (p.23) wasomitted, garbling the meaning of the content. [Note These corrections have been incorporatedinto the text in this electronic version – TIS]