In the thread 'Constructive Interference and Radiowave Propagation', Owen, on 4-8-07 asserted that my writings
in Reflections concerning the analysis of stub matching procedures using reflection coefficients are
applicable only in cases where the transmission line is either lossless, or distortionless. I disagree, and in
what follows I hope to persuade those who agree with Owen's position to reconsider.

To assist in understanding why my use of reflection coefficients in analyzing impedance-matching circuitry, I
find it useful to include the concept of virtual open- and short-circuit conditions. I realize that some of
the posters on this NB deny the existence of virtual open-and short-circuits. Therefore, I hope that my
presentation here will also persuade those posters to reconsider their position.

While working in an antenna lab for more than 50 years I have analyzed, constructed, and measured hundreds of
impedance-matching circuits comprising transmission-line circuitry using reflection coefficients as
parameters. For example, in 1958 my assignment was to develop the antenna system for the World's first weather
satellite, TIROS 1. The system required an antenna that would radiate efficiently on four different
frequencies in two bands that were more than an octave related. It required a coupling circuit that would
allow four transmitters to operate simultaneously on all four frequencies without mutual interference. After
developing the antenna that also required radiating circular polarization, I then developed the coupling
system, which, pardon my English, utilized several virtual open- and short-circuit conditions to accomplish
the required isolation between the individual transmitters. The entire coupling system was fabricated in
printed-circuit stripline transmission line (not microstrip), with no connectors except for transmitter input
ports and output ports feeding the antenna. Remember, this was in 1958.

Initially I had only a slotted line for impedance measurements during the development stage, but soon after
the PRD-219 Reflectometer became available, invented by my bench mate, Woody Woodward. The PRD-219 measured
SWR and the angle of the voltage reflection coefficient. The magnitude rho of the reflection coefficient was
obtained from the SWR measurement using the equation rho = (SWR - 1)/(SWR + 1), thus the PRD actually measured
the complete complex reflection coefficient. Consequently, all measurements from then on were in terms of
reflection coefficient.

Keep in mind that I was working with real transmission lines--not lossless lines. There were several
stub-matching circuits, several occurrences of virtual open- and short-circuits, and the total loss through
the coupler at both the 108 and 235 MHz bands was no greater than 0.2 dB. The input SWR at all four input
ports for a run of 12 manufactured units never exceeded 1.05:1 relative to 50 ohms.

Please let me now explain my understanding of virtual open- and short-circuits. These circuits are developed
by interference between two sets of voltage and current waves having reflection coefficients of equal
magnitude and phase differences of 180°, respectively. Consider these two examples of a virtual short circuit:

1: The input impedance of a lossless half-wave (180°) transmission line terminated in a physical short circuit
is zero ohms, a short circuit, but a VIRTUAL short circuit because it was achieved only by the interference
between the source voltage wave incident on the input (0°) and the reflected voltage wave (180°) returning to
the input after 360° of two-way travel on the line and the 180° phase reversal at the physical short
terminating the line. The reflected current wave on return to the input encountered no phase change during its
travel, thus the current reflection coefficient is in phase with that of the source current, allowing the
short circuit to occur.

2: The input impedance of a lossless quarter-wave (90°) transmission line terminated in a physical open
circuit is zero ohms, a short circuit, but a VIRTUAL short circuit because it was achieved only by the
interference between the voltage wave incident on the input (0°) and the reflected voltage wave (180°)
returning to the input after 180° of two-way travel on the line and the 0° phase reversal at the physical open
circuit terminating the line. The current reflection coefficient occurs in the same manner as with the
half-wave line above.

These two examples can be confirmed by referring to any reputable text concerning transmission line theory.

The voltage reflection coefficient at the input of these two transmission lines is 1.0 at 180°, and the
current reflection coefficient at this point is 1.0 at 0°. These are the reflection coefficients that would be
found when measuring at any short circuit, no matter whether it is physical or virtual. Consequently, both
physical and virtual short or open circuits placed on a transmission line can cause reflections. Proof is in
measurements performed at various points in the antenna coupler developed for the TIROS spacecraft in 1958.

Now let's examine a specific example of impedance matching with a stub using reflection coefficients, with
more details than I used in the previously-mentioned thread. As I said earlier, I have measured hundreds of
stub-matching circuitry, but for this discussion, yesterday I set up an experimental stub-matching circuit for
the purpose of being able to report directly on the results of current measurements taken on the circuit. The
source is an HP-8640A signal generator, an HP-5328A counter to determine the operating frequency, and the
combination of an HP-8405 Vector Voltmeter and an HP-778D dual directional coupler to form a precision RF
network analyzer.

Because using a 3:1 mismatch the resulting numbers are convenient, I paralleled three precision 50-ohm
resistors to form a resistance of 16.667 ohms, resulting in a 3:1 mismatch on the line to be stubbed. On a
line with a 3:1 mismatch the correct positioning of a parallel matching stub is 30° toward the source from a
position of minimum SWR, where the normalized admittance y = 1.0 + 1.1547. Thus, I selected a short piece of
RG-53 coax that measured exactly 30° in length at 16.0 MHz, meaning the stub will be placed 30° rearward of
the load.

All measurements obtained during the experiment were less than 2 percent in error compared to a perfect
text-book setup. Consequently, rather than bore you with the exact measured values, I'm going to use the
text-book values for easier understanding.

At the 16.667 + j0 load the measured voltage reflection coefficient = 0.5 at 180°, current 0.5 at 0°.
At the stub point voltage reflection coefficient of the line impedance = 0.5 at +120°, current 0.5 at -60°.
Open-circuited stub 49° in length measured separately in parallel with 50 ohms yields voltage reflection
coefficient 0.5 at -120°, current 0.5 at +60°. (Keep in mind that in operation the stub is in parallel with
the 50-ohm line resistance at the stub point.)
With stub connected in parallel with the line the voltage reflection coefficient at the stub point is 0.04 at
0°, current 0.04 at 180°. (Equivalent SWR = 1.083, and impedance = 54.16 + j0 ohms.)

Summarizing reflection coefficient values at stub point with stub in place:
Line coefficients: voltage 0.5 at +120°, current -60°
Stub coefficients: voltage 0.5 at -120°, current +60°
Resultant coefficients: voltage 0.5 at 180°, current 0.5 at 0°

These two resultant reflection coefficients resulting from the interference between the load-reflected wave at
the stub point and the reflected wave produced by the stub define a virtual short circuit established at the
stub point.

Let's now consider what occurs when a wave encounters a short circuit. We know that the voltage wave
encounters a phase change of 180°, while the current wave encounters zero change in phase. Note that the
resultant voltage is at 180°, so the voltage phase changes to 0° on reflection at the short circuit, and is
now in phase with the source voltage wave. In addition, the resultant current is already at 0°, and because
the current phase does not change on reflection at the short circuit, it remains at 0° and in phase with
source current wave. Consequently, the reflected waves add in phase with the source waves, thus increasing the
forward power in the line section between the stub and the load.

So how do we know that the virtual short circuit resulting from the interference is really performing as a
short circuit?

First, an insignificant portion of the reflected wave appears on the source side of the stub point, thus, from
a practical viewpoint, indicating total re-reflection of the reflected waves at the stub point.

Second, the voltage in the line section between the stub and load that has a 3:1 SWR has increased relative to
that on the source line by the factor 1.1547, the amount expected on a line having a 3:1 SWR after total
re-reflection at an open or short circuit. This increase factor is determined from the equation for the
increase in forward power on a line with a specific value of SWR, where rho is the corresponding value of
reflection coefficient. The power increase factor equation is power increase = 1/(1 - rho^2). Thus the voltage
increase factor is the square root of the power increase factor. With rho = 0.5, as in the case of the above
experiment, the power increase factor is 1.3333..., the square root of which is 1.1547.

We have thus proved that the virtual short circuit established at the stub point is actually performing as a
real short circuit.

I believe it is remarkable that the maximum deviation of the measured values obtained during the experiment is
less than 2 percent of the text-book values that would appear with lossless elements, and ignoring measurement
errors and tolerances of the measuring equipment. The recognized sources of error a
1. Tolerance in readings from the Vector Voltmeter
2. Ripple in the coupling factor in the directional coupler
3. Attenuation in the coax
4. The fact that the nomional Zo of the RG-53 coax is 53.5 ohms, not 50, as used as the reference in the
measurements.

My final comment is that I hope I have assisted in appreciating the practical use of virtual open and short
circuits, and that matching procedures can be analyzed using reflection coefficients that are not restricted
to lossless or distortionless transmission lines.

Walt, W2DU