A Feedline Energy Analysis

First published in WorldRadio, Oct. 2005, and reproduced with permission.

An Energy Analysis at an Impedance Discontinuity in an RF Transmission Line, Part I

Where Does the Power Go? [1] There continue to be many differing responses to the question within the amateur radio community and, so far, no one has presented the facts of the physics of power as understood from the field of optics. Those facts from optics have been known and understood for decades and are consistent with the laws of physics and the equations governing the behavior of RF transmission lines. Light and RF waves are both composed of electromagnetic energy. Most of the following information comes from "Optics" [2]. In the field of optics, irradiance is the same thing as power in an RF transmission line if the cross sectional area of the transmission line is taken into account. Irradiance has the dimensions of energy per unit area per unit time. If the light beam of a particular laser occupies the same cross sectional area as a particular coaxial RF transmission line then the irradiance of the laser beam is comparable to the RF power in the transmission line. The 1/4 wavelength thin-film deposited on glass to obtain a non-reflective surface performs in a virtually identical way to a 1/4 wavelength series matching section in a transmission line. Single-source RF energy in a transmission line and laser light are both coherent electromagnetic energy waves that obey the laws of superposition, interference, conservation of energy, and conservation of momentum.

My Historical Perspective

My first memories of the answer to "Where does the power go?" are articles published in QST written by Walter Maxwell, W2DU, some quarter of a century ago. Mr. Maxwell later compiled the information into a book titled, "Reflections", which quickly became the bible for Amateur Radio applications involving stub matching, transmission lines, and forward and reflected energy flow. Mr. Maxwell coined the terms, "virtual short" and "virtual open", as a shorthand description of what rearward-traveling reflected energy encounters at a match point in a transmission line resulting in 100% re-reflection. He also explained the function of destructive wave interference and constructive wave interference in achieving a match point on a transmission line [8] which is what a large part of this article is about.

Sometime after the publication of Reflections, some people questioned the validity of Mr. Maxwell's concepts. In particular, Dr. Steven Best, VE9SRB, took Mr. Maxwell to task in a series of articles published in QEX [3]. Simply put, Dr. Best disagreed with Mr. Maxwell that reflected power is 100% re-reflected in a matched system. Before publication of his Part 3 QEX article, Dr. Best sent up trial balloons for his ideas on the usenet newsgroup, rec.radio.amateur.antenna. My opinion was that Dr. Best's future article contained numerous errors which were pointed out to him. However, the article as published still contained the alleged errors. My determination to resolve the conflicts between the concepts presented by Walter Maxwell and the ones presented by Dr. Best culminated in this present article. The conclusions will be presented first with the technical details to follow in Part II and Part III.

In a nutshell, Walter Maxwell's "virtual short" is a two step process. The reflected wave from the load encounters the impedance discontinuity at the match point. A re-reflection occurs that equals the incident reflected power multiplied by the power reflection coefficient at the match point (the square of the voltage reflection coefficient). This re-reflected energy joins the forward wave traveling toward the load. That first energy re-reflection is not the only energy that joins the forward wave. That fact is what Dr. Best missed in his article. Interference of any kind was never mentioned in Dr. Best's QEX article.

The part of the reflected wave that is not re-reflected is transmitted back through the impedance discontinuity at the match point and attempts to flow toward the source. We know the reflected energy doesn't make it to the source in a matched system, so where does it go? The answer is mentioned in "Reflections II" [8]. What Mr. Maxwell is describing is wave cancellation due to total destructive interference between two reflected waves. The first wave is the part of the source forward wave that is initially reflected back toward the source from the match point. The second wave is the part of the reflected wave from the load that is transmitted through the match point toward the source. These waves are equal in magnitude and opposite in phase so, as Mr. Maxwell asserts in Reflections II, they cancel to zero at the match point thus eliminating reflections between the match point and the source.. The canceling of these two waves to zero is the second step in Mr. Maxwell's virtual short process of 100% reflection.

Voltages can cancel and currents can cancel but energy cannot cancel. What happens to the energy that existed in the waves before they were cancelled? Since we know that all the energy in a matched system winds up flowing toward the load, the answer is a no-brainer. There are only two directions in a transmission line. If energy that was previously flowing toward the source isn't flowing toward the source anymore, it must necessarily be flowing toward the load. The conclusion is inescapable. Not only is 100% of the reflected energy re-reflected at the match point, but wave cancellation is the cause of part of that re-reflection. This is a well understood phenomenon in the field of optics [9] but not well understood in the field of RF engineering.

An RF engineer will usually say there are three things that can cause 100% reflection. Those are a short-circuit, an open-circuit, or a purely reactive impedance. This is true at a load. But at an impedance discontinuity with waves incident from both directions, to that list of three, we can add a fourth, namely wave cancellation due to total destructive interference. In general:

The destructive interference energy resulting from wave cancellation at an impedance discontinuity becomes an equal magnitude of constructive interference in the opposite direction. Since there are only two directions in a transmission line, wave cancellation is the equivalent of an energy reflection. 100% wave cancellation means 100% energy reflection. [9]

References are included in this Part I of the series and will not be repeated in Parts II and III. In Part II, the general case qualitative analysis will be presented.

I would like to thank Mr. Robert E. Lay, W9DMK, for his substantial contributions to this article.

References

[1] Bloom, Jon, Where Does the Power Go?, "QEX", Dec. 1994

[2] Hecht, Eugene, "Optics", Fourth Edition, (c)Aug. 2001, Addison-Wesley, ISBN 0805385665

[3] Best, Steven R., Wave Mechanics of Transmission Lines, Part 3, "QEX", Nov/Dec 2001

[4] "Interference term", "Optics", Eugene Hecht, Fourth Edition

Section 7.1 The Addition of Waves of the Same Frequency It follows ... that the resultant flux density is not simply the sum of the component flux densities; there is an additional contribution 2*E01*E02*cos(a2-a1), known as the interference term. ("a" replaces the Greek letter Alpha and INTERFERENCE TERM is emphasized.)

Section 9.1 General Considerations
The 'interference term' becomes I12 = 2*SQRT[(I1)(I2)]*cos(s) [where I is irradiance (power)] ('SQRT' replaces the square root sign and "s" replaces the Greek letter Sigma.)

[5] "S-Parameter Techniques", Hewlett Packard Application Note 95-1, available on the web. The S-Parameter normalized voltage equations are:

b1 = (s11)(a1) + (s12)(a2) and b2 = (s21)(a1) + (s22)(a2)

The squares of all those terms are related to power as explained in the application note. It is left as an exercise for the reader to square both sides of both equations above and observe that the resulting equations contain the interference term that agrees with Eq 1 and Eq 2 in the body of this paper.

[6] "Optics", Eugene Hecht, Fourth Edition

Section 3.3 Energy and Momentum, "One of the most significant properties of the electromagnetic wave is that it transports energy and momentum." [Note from W5DXP: Energy and momentum must be conserved. The direction of the energy and momentum associated with reflected waves must be reversed for a match to occur.]

Section 4.11 Photons, Waves and Probability, "The principle of conservation of energy makes it clear that if there is constructive interference at one point, the 'extra' energy at that location must have come from somewhere else. There must therefore be destructive interference somewhere else. "If two or more electromagnetic waves arrive at point P out-of-phase and cancel, 'What does that mean as far as their energy is concerned?' Energy can be distributed, but it doesn't cancel out."

Section 7.1 The Addition of Waves of the Same Frequency, "The superposition of coherent waves generally has the effect of altering the spatial distribution of the energy but not the total amount (of energy) present."

[7] "Optics", Eugene Hecht, Fourth Edition

Section 9.1 General Considerations, "A maximum irradiance (power) is obtained when cos(s) = 1. ... In this case of total constructive interference, the phase difference between the two waves is an integer multiple of 2*Pi, and the disturbances are in-phase. ... A minimum irradiance (power) results when the waves are 180 degrees out-of-phase, ... cos(s) = -1, ... and is referred to as total destructive interference." ("s" replaces the Greek letter Sigma and TOTAL DESTRUCTIVE INTERFERENCE is emphasized.)

[8] Maxwell, Walter, "Reflections II", (c) 2001 Worldradio Books, ISBN 0-9705206-0-3 page 4-3, "The destructive wave interference between these two complementary waves ... causes a complete cancellation of energy flow in the direction toward the generator. Conversely, the constructive wave interference produces an energy maximum in the direction toward the load, ..." page 23-9, "Consequently, all corresponding voltage and current phasors are 180 degrees out of phase at the matching point. ... With equal magnitudes and opposite phase at the same point (point A, the matching point), the sum of the two (reflected) waves is zero."

[9] Quotes from two web pages from the field of optical engineering:

www.mellesgriot.com/products/optics/oc_2_1.htm

"Clearly, if the wavelength of the incident light and the thickness of the film are such that a phase difference exists between reflections of p, then reflected wavefronts interfere destructively, and overall reflected intensity is a minimum. If the two reflections are of equal amplitude, then this amplitude (and hence intensity) minimum will be zero." (Referring to 1/4 wavelength thin films.)

"In the absence of absorption or scatter, the principle of conservation of energy indicates all 'lost' reflected intensity will appear as enhanced intensity in the transmitted beam. The sum of the reflected and transmitted beam intensities is always equal to the incident intensity. This important fact has been confirmed experimentally."

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/interference/waveinteractions/index.html

"... when two waves of equal amplitude and wavelength that are 180-degrees ... out of phase with each other meet, they are not actually annihilated, ... All of the photon energy present in these waves must somehow be recovered or redistributed in a new direction, according to the law of energy conservation ... Instead, upon meeting, the photons are redistributed to regions that permit constructive interference, so the effect should be considered as a redistribution of light waves and photon energy rather than the spontaneous construction or destruction of light."

Note from W5DXP: In an RF transmission line, since there are only two possible directions, the only "regions that permit constructive interference" at an impedance discontinuity is the opposite direction from the direction of destructive interference.

Cecil A. Moore, w5dxp@arrl.net

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