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,

**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,

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

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

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 ZØ_{1}. 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 ZØ_{2}.

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.

P_{3} = P_{ref2}(1 - ρ^{2}) and P_{4} = P_{for1}(ρ^{2})

P_{1} + P_{4} = P_{for1} and P_{2} + P_{3} = P_{ref2}

The following equations are paraphrased from similar relationships published in ** Optics**. The symbol
'I' representing

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 V

If ( 0 ≤ θ < 90 ) then there exists constructive interference between V_{1} and V_{2}, i.e. cos(θ) is a
positive value. Therefore there exists an equal magnitude of destructive interference between V_{3} and
V_{4} 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 V_{1} and V_{2}.
There is also no destructive/constructive interference between V_{3} and V_{4}. 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 V_{1} and V_{2}, i.e. cos(θ) is a
negative value. Therefore there exists an equal magnitude of constructive interference between V_{3} and
V_{4} 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 V_{2} is leading or lagging
V_{1} 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 V _{1} and V_{2} is always zero
degrees (ZØ_{2} > ZØ_{1}) or 180 degrees (ZØ_{2} < ZØ_{1}).**

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 V_{1} and V_{2} is zero degrees for total constructive interference. The phase angle between
V_{3} and V_{4} 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 P_{3} to equal P_{4}. From that fact and knowing that (P_{1})(P_{2}) = (P_{3})(P_{4}), it can be
shown that, in a matched system:

|P_{1}| + |P_{4}| = |P_{for1}| and |P_{2}| + |P_{3}| = |P_{ref2}|

We can conclude that: |P_{for2}| = |P_{for1}| + |P_{ref2}|

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. P_{2} = P_{ref2} ( ρ^{2} ) is the first re-reflection event and occurs when the reflected wave associated
with P_{ref2} encounters the impedance discontinuity.

2. P_{3} and P_{4} are associated with two waves involved in total destructive interference. Since the related voltages, V_{3} and V_{4},
are equal in magnitude and 180 degrees out of phase, the energy components in P_{3} and P_{4} cease to exist
on the source side of the impedance discontinuity, and instead are redistributed toward the load as total constructive
interference. P_{3} and P_{4} 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
P_{3} and P_{4} 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. P

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 P_{3} wave can be inferred from P_{3} = P_{ref2}(1- ρ^{2}) where P_{ref2} and
(1-ρ^{2}) do exist. Given that the P_{3} wave exists, then the existence of the P_{4} 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 <--Pref2The source is supplying 100 watts. The SWR meter reads 1:1 on the coax. With the information given, can we calculate the forward power, reflected power, and SWR on the twinlead? How about voltages and currents on the twinlead?

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 2E_{01}E_{02}cos(α_{1}-α_{2}), known as ** the interference term**.

Section 9.1 General Considerations: The 'interference term' becomes I_{12} = 2*SQRT[(I_{1})(I_{2})]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

[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.