An Energy Analysis at an Impedance Discontinuity in an RF Transmission Line

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 else has presented the facts of the physics of electromagnetic energy as understood from the field of optics. Those technical 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 the power flow vector in an RF transmission line. Irradiance, like a power flow vector, has the dimensions of energy per unit time per unit area. 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. The power density terms in the irradiance equation have been multiplied by the unit area of the transmission line to obtain the resulting power equation in watts.

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 [10]. 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.

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, the two wavefronts 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% "re-reflection" (actually reflection plus redistribution [10]).

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 redistributed back toward the load at the match point, but wave cancellation is the cause of part of that redistribution. This is a well understood phenomenon in the field of optics [9] but not well understood in the field of RF engineering.[10]

An RF engineer will tell us that there are three things that can cause 100% reflection. Those are a short-circuit, an open-circuit, or a purely reactive impedance. But there is another phenomenon that can cause the reflected energy to reverse direction and flow toward the load - wave cancellation. 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 redistributes the energy in the opposite direction. [9] This redistributed energy joins the forward wave just as the re-reflected energy does.

The General Case Qualitative Analysis

The following discussion of a generalized impedance discontinuity in an RF transmission line, operating under steady-state conditions, is not intended to replace a conventional quantitative voltage analysis. It is intended instead as a conceptual qualitative energy analysis that extends the voltage analysis and answers that original question: Where Does the Power Go?

The following diagram is of a generalized impedance discontinuity in a typical RF transmission line. P_for1 is the forward power and P_ref1 is the reflected power on the source side section of transmission line having a characteristic impedance of ZØ₁. Likewise, P_for2 is the forward power and P_ref2 is the reflected power on the load side section of transmission line having a characteristic impedance of ZØ₂.

The following equations govern the distribution of energy at the impedance discontinuity, which, of course, is constrained by the conservation of energy principle. The Greek letter, 'ρ', will be used to designate the voltage reflection coefficient.

While the voltage reflection coefficients are of opposite signs on the two sides of the junction, the power reflection coefficient is always positive on either side of the junction. The same concept applies to the power transmission coefficient. Since the impedance discontinuity is an imaginary plane and contributes no losses, all the power that is not reflected at the impedance discontinuity plane is transmitted through the impedance discontinuity plane.

The following equations are paraphrased from similar relationships published in Optics. The symbol 'I' representing irradiance in a light wave has been replaced by the symbol 'P' representing the power flow vector in an RF wave (or RF wave component) in a transmission line. Note that these equations apply to coherent waves. V₁ and V₂ are superposed at the load side of the impedance discontinuity, using phasor math, to obtain V_for2 where 'θ' is the relative phase angle between V₁ and V₂. Note that cos(θ) = cos(-θ), so for an energy analysis, it doesn't matter if V₂ is leading or lagging V₁.

The following is the key equation that Dr. Best neglected to include in his QEX article. It is the other half of the energy equation and brings all of the energy into balance as required by the conservation of energy principle. It can be shown that the relative phase angle between V₃ and V₄ is (180 degrees minus θ) and since cos(θ) = -cos(180-θ), the following equation applies when V₃ and V₄ are superposed at the source side at the impedance discontinuity:

If ( 0 ≤ θ < 90 ) then there exists constructive interference between V₁ and V₂, i.e. cos(θ) is a positive value. Therefore there exists an equal magnitude of destructive interference between V₃ and V₄ where cos(180-θ) is a negative value. A positive sign on the interference term indicates constructive interference. A negative sign on the interference term indicates destructive interference.

If θ = 90 deg, then cos(θ) = 0, and there is no destructive/constructive interference between V₁ and V₂. There is also no destructive/constructive interference between V₃ and V₄. Any potential destructive/constructive interference between any two voltages is eliminated because θ = 90 deg, i.e. the voltages are superposed orthogonal to each other (almost as if they were not coherent).

If (90 < θ ≤ 180) then there exists destructive interference between V₁ and V₂, i.e. cos(θ) is a negative value. Therefore there exists an equal magnitude of constructive interference between V₃ and V₄ where cos(180-θ) is a positive value.

One note of importance is that, in the case of a mismatched impedance discontinuity, reflected power is not 100% re-reflected and redistributed. Dr. Best was right about that.

Now we are in a position to discover something that falls out from the conservation of energy principle. A simple mathematical manipulation of equation 3 above will show that:

Why not turn this qualitative analysis into a quantitative analysis? If we know the forward power and reflected power on each side of the impedance discontinuity and the reflection coefficient at the impedance discontinuity, we can certainly do a quantitative analysis. The only problem is that there are two solutions. Without additional information, one cannot tell whether V₂ is leading or lagging V₁ and therefore there exists two possible solutions. In order to confine the results to one unique solution, one would need to know the number of wavelengths between the discontinuity and the load and the reflection coefficient at the load. But as we shall discover in Part III: For a ZØ-matched system, the two-solution problem disappears because the phase angle between V₁ and V₂ is always zero degrees (ZØ₂ > ZØ₁) or 180 degrees (ZØ₂ < ZØ₁).

Note: It cannot be over-emphasized that "wave cancellation" does not imply energy cancellation. "Wave cancellation" refers to the cancellation of two coherent EM voltage/current waves traveling the same path in the same direction. The energy components in the cancelled waves cannot be destroyed so the energy must seek another path, i.e. it is redistributed in the opposite direction in a transmission line.

The Special Case ZØ-Matched Analysis

Someone might ask, why bother with a special case? As it happens, this special case applies to all matched systems for which P_ref1 = 0, and is the most common case within amateur radio. One might say the matched case is not all that 'special'. Here's a diagram for the ZØ-matched system.

When two coherent waves of equal magnitude and opposite phase encounter each other while moving in the same direction in a transmission line, complete cancellation of the two waves occurs. The total voltage goes to zero and the total current goes to zero in the direction of the canceled waves. V_ref1 (and I_ref1) go to zero at the match point. That's entirely logical since the reflected power flow vector toward the source equals zero in a matched system.

In a ZØ-matched system with reflections, total destructive interference occurs at the source side of the ZØ-match point and eliminates reflections toward the source. Following the principle of conservation of energy, the destructive interference energy previously associated with the two cancelled reflected waves becomes total constructive interference energy flowing toward the load as part of P_ref2.

The energy equations governing the behavior of a ZØ-matched system are simplified because the phase angle between V₁ and V₂ is zero degrees for total constructive interference. The phase angle between V₃ and V₄ is 180 degrees for total destructive interference.

For a ZØ-matched system, in which all reflections are cancelled toward the source, it is necessary for P₃ to equal P₄. From that fact and knowing that (P₁)(P₂) = (P₃)(P₄), it can be shown that, in a matched system:

Which is what a lot of people have been saying for a lot of years.[8] Note: The absolute value marks are included to indicate that these powers are scalar values, not power flow vectors.

Conclusion: There are two steps leading to the total redistribution of reflected energy back toward the load in a ZØ-matched system.

1. P₂ = P_ref2 ( ρ² ) is the first re-reflection event and occurs when the reflected wave associated with P_ref2 encounters the impedance discontinuity.

2. P₃ and P₄ are associated with two waves involved in total destructive interference. Since the related voltages, V₃ and V₄, are equal in magnitude and 180 degrees out of phase, the energy components in P₃ and P₄ cease to exist on the source side of the impedance discontinuity, and instead are redistributed toward the load as total constructive interference. P₃ and P₄ are power flow vectors associated with two reflected waves that cancel, so according to the principle of conservation of energy, their combined energy (and momentum) must change direction. Since P₃ and P₄ are reflected energy components that end up flowing toward the load, the interference event can be considered to be a redistribution [9] of reflected energy. Combining steps 1 and 2 above, it is apparent that 100% of the reflected energy is re-reflected and redistributed in a matched system. P₂, P₃, and P₄ are all reflected wave components.

Note that the author previously used the word "reflection" for both actions involving a single wave and the interaction between two waves. Now the word "reflected" is being used only for single waves and the word "redistributed" is being used for the two wave interference scenario.

Step 2 above, is a somewhat new concept in the field of RF engineering although it has existed for decades in the field of optics.[9] We amateur radio operators can add an item to the list of things that can cause a redistribution of the energy in incident waves: 1. Short-circuit, 2. Open-circuit, 3. Pure reactance, 4. Wave cancellation.[10] Wave cancellation cannot occur at a single load fed by a single source through a single transmission line but it can occur in a transmission line when waves are incident upon an impedance discontinuity from both directions. Although harder to understand and prove, wave cancellation (and therefore 100% redistribution of canceled wave energy) can also occur at (or inside) a source when RF energy is coming from both directions. For instance, reflected wave cancellation will occur in a tube-type final amp when the pi-network is tuned for system resonance. In that case, the reflected wave cancellation point would be the Zg-match point inside the transmitter.

Note: The steady-state existence of the P₃ wave can be inferred from P₃ = P_ref2(1- ρ²) where P_ref2 and (1-ρ²) do exist. Given that the P₃ wave exists, then the existence of the P₄ wave is necessary for wave cancellation. Unfortunately, superposition happens faster than humans can observe, even with their fastest instruments.

And that is where the power goes; (actually, it is the energy that does the "going").

A Simple Example

Consider the following lossless system with a 1:1 choke at point 'x':

100W XMTR---50 ohm coax---x---300 ohm twinlead---load
		Pfor1-->	Pfor2-->
		<--Pref1	<--Pref2

The power reflection coefficient is [(300 - 50)/(300 + 50)]² = 0.51 and the power transmission coefficient is (1 - 0.51) = 0.49. For the system to be matched, these coefficients must also exist at the load. So the forward power on the twinlead must be 100W/0.49 = 204.1 watts. That makes the reflected power on the twinlead equal to (204.1 - 100) = 104.1 watts. From these two power values, we can calculate V_for2 = 247.4V, I_for2 = 0.825A, V_ref2 = 176.7V, I_ref2 = 0.589A , SWR(300) = 6:1

The SWR can be calculated in any number of ways. VSWR(300) = (V_for2 + V_ref2)/(V_for2 - V_ref2)

If we know the physical length and velocity factor of the 300 ohm twinlead, we can actually calculate the feedpoint impedance of the load (antenna).

The author has endeavored to satisfy the purists in this series of articles. The term "power flow" has been avoided in favor of "energy flow". Power is a measure of that energy flow per unit time through a plane. Likewise, the EM fields in the waves do the interfering. Powers, treated as scalars, are incapable of interference. Any sign associated with a power in this paper is the sign of the cosine of the phase angle between two voltage phasors. A plus sign indicates constructive interference (or energy flow toward the load) and a minus sign indicates destructive interference (or energy flow toward the source).

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 2E₀₁E₀₂cos(α₁-α₂), known as the interference term.

Section 9.1 General Considerations: The 'interference term' becomes I₁₂ = 2*SQRT[(I₁)(I₂)]cos(σ)
(where 'SQRT' replaces the square root sign.)

[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(σ) = 1. ... In this case of total constructive interference, the phase difference between the two waves is an integer multiple of 2π, and the disturbances are in-phase. ... A minimum irradiance (power) results when the waves are 180 degrees out-of-phase, ... cos(σ) = -1, ... and is referred to as total destructive interference."

[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."

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.

[10] Revision 1.1, Feb. 20, 2008 - In the original version, the redistribution of energy due to wave cancellation was called a "reflection", a common practice in amateur radio circles. W5DXP has changed that description in favor of a "redistribution" as described by the FSU web page. The word "reflection" is reserved for describing the event when a single wave encounters an impedance discontinuity. This is accordance with The IEEE Dictionary definition of "reflected wave". The word "redistribution" of energy is adopted for describing what happens to the energy when two or more waves interact. In like manner, since interference can occur with or without permanent wave interaction, interference alone is necessary but not sufficient to correspond to the permanent redistribution of energy.