[TowerTalk] Reflections, Tuners and Antenna Forum Talks

[Steven Best]

This post is about a day late.  It is a modified copy of a message I sent
Jim regarding his power measurements.  Since Jim received this message, he
has made additional measurements, posts and noted a few changes in his Z
measurements. (Jim, this message has been changed to reflect the new Z you
gave me and your new measurements.  The overall discussion has been modified
a bit as well.)

This is LONG!

Very good experiment.  I really have no issues with the set up, procedure,
or the steady state results.  I would like to put them in perspective
relative to my e-mail yesterday/today regarding the relationships between
voltage, power and wave reflections.

I will skip right to the results with the external Palstar tuner.  As I
understand it, Jim had a steady state input impedance at the shack end of
the cable of Zin = 17 + j11 ohms.  For the sake of discussion, I assumed
that the reactive component of the impedance has a + sign.  The operating
frequency is 7.088 MHz.  Jim then adjusted the tuner such that it had a
steady state input impedance of near 50 ohms (50 + j8 ohms).  Jim then
delivered 1000 watts of forward power to the tuner input.  The reflected
power was measured at about 8 to 10 watts.    At the tuner output, Jim
measured about 1150 - 1200 watts of forward power (with a possible error of
+ - 125 watts) and 230 - 300 watts of reflected power (with a possible error
of + - 25 watts).  Given these results, it would appear that the approximate
effective net power delivered to the transmission line is between 850 - 970
watts (depending on how you run the numbers).

In order to run an analysis for discussion, I had to make a lot of
assumptions.  I will assume that the 1000 watts of forward power at the
tuner input is correct.  I will assume that the forward power at the tuner
output is 1175 watts and that the reflected power is 265 watts.  This would
result in an effective net power of 910 watts delivered to the transmission
line and 90 watts of power being dissipated in the tuner (tuner loss is the
critical number for my analysis).  These numbers are probably not correct
but this is really not relevant to the discussion that follows.  Now let's
interpret Jim's measurements re: wave reflections.

Based upon Walt's method of analysis, he would indicate that 910 watts of
power is delivered to the transmission line connected to the antenna (the
effective net power delivered to the transmission line).  The reflected
power at the tuner output is 265 watts and the total forward power at the
tuner output is 1175 watts.  The forward power is a result of the 265 watts
of reflected power being totally re-reflected in-phase with the 910 watts of
power and the two add to become the 1175 watts of total forward power.
Looking no further into these steady state conditions this appears to make
sense and all seems to be okay.

Now let's look at the voltage analysis.  1000 watts of forward power MUST be
associated with a forward voltage having a magnitude of 223.6 volts
(assuming Zo is 50 ohms).  265 watts of forward power MUST be associated
with a forward voltage having a magnitude of 115.1 volts.  If these voltages
add "in-phase", then the resulting forward power MUST be 2294 watts NOT 1175
watts.  So then, looking at Walt's analysis of the steady state conditions,
why does there appear to be an inconsistency in the relationship between
voltage and power??

Note also that the total re-reflection of reflected power must require that
the level of re-reflection is entirely independent of the impedance
terminating the transmission line.  Here is another inconsistency in Walt's
analysis in that general transmission line theory demands that wave
reflections at either end of a transmission line are a direct result of the
impedance terminating the line.  From the reflected wave's perspective, the
impedance "terminating" the transmission line is the output impedance of the
tuner, which, by my definition is the impedance measured or seen looking
rearward into the tuner's output terminals.

At this point, one would ask is there another method of analysis that yields
the correct steady state conditions AND bases reflections at either end of
the transmission line on the impedance terminating the line AND develops
relationships between voltage and power that are consistent with circuit
theory?  The answer is yes.

First, we need a brief review of Transmission line theory 101.  Let's assume
that we have a transmitter connected to a transmission line terminated with
an antenna having an impedance Za not equal to Zo.  When the transmitter is
first energized, what impedance does it see at the input to the transmission
line?  The answer is Zo.  Since there are no reflections in the system at
this point, the only impedance the transmitter sees is Zo.  For this reason,
the driving forward voltage, current and power the transmitter delivers to
the transmission line is equal to the voltage, current and power it would
deliver to a load of Zo. Lets define the transmitter's initial forward
driving voltage to be Vi.

In the steady state, what impedance does the transmitter see?  This is a
tricky question.  The obvious answer is Zin but what does this really mean?
>From the initial state to the steady state, what physical impedances have
changed in the system?  The answer is none.  Za does not change and Zo does
not change.  So then, why does the input impedance to the transmission line
change?  The input impedance of the transmission line changes only as a
result of voltage and current reflections arriving back at the transmission
line input (transmitter output).  At the same time, there will be some level
of re-reflected voltage and current developed at the transmitter output.
The input impedance at the transmission line input is not a physical
impedance.  It is simply the ratio of total voltage to total current at the
transmission line input.  Lets define the reflected voltage arriving at the
transmitter output as Vr and the re-reflected voltage developed at the
transmitter output as Vrer.

Now one of the most important questions or concepts to consider.  Does the
forward driving voltage, current and power "delivered" by the transmitter
really change as the system transitions to the steady state?  The answer is
no but this is a difficult concept for some people to accept.  Changes in
input impedance at the transmitter output are really not physical changes in
load impedance.  All the transmitter sees is a rearward traveling voltage
source developing at its output terminals.  This rearward voltage source
developing at the transmitter's output terminals causes the internal voltage
and current within the transmitter to change.  There is no physical change
in load impedance.  Yes, the ratio of total voltage to total current changes
but this is not quite the same as a physical change in load impedance.  The
forward voltage, current and power delivered to the transmission line also
change but not as a result of the transmitter's source parameters changing.
These change only because the re-reflected voltage and current add to the
initial forward driving voltage and current delivered by the transmitter.
>From the perspective of the transmitter's internal circuitry, the steady
state condition can be represented with a forward driving source (the
transmitter's forward source) and a rearward driving source (the sum of the
reflected and re-reflected voltages at the transmitter output).  The steady
state internal voltage, current and power developed within the transmitter
are the net result of the forward driving voltage and current MINUS the
rearward driving voltage and current.  This has the appearance of the
transmitter adjusting its "delivered" voltage and current to compensate for
reflections but that's not quite true.  This may all be a matter of
interpretation and this topic probably requires a separate discussion so
let's move forward at this point.

The important issue is what does this mean for the wave reflection analysis?
The total voltage developed at the transmitter output is equal to Vi + Vr +
Vrer.  The total rearward voltage delivered back into the transmitter is Vr
+ Vrer.  The total forward traveling voltage delivered to the transmission
line is Vi + Vrer.  Critically important is the fact that Vrer is equal to
Vr multiplied the by the reflection coefficient of the transmitter's output
impedance.

Now back to Jim's tuner experiment.  I ran an analysis of the set up Jim
described, which is the basis for the numbers presented here.  They are not
100% accurate since I don't know the exact tuner loss.  The assumed tuner
loss is a critical assumption for the analysis that I present here.  My
assumptions should be close enough though to establish the conceptual and
mathematical relationships between wave reflections, power and voltage.

I assumed that the 1000 watts of forward power arriving at the tuner input
is simply a voltage of 223.6 volts (0 degrees relative phase).  I assumed
that the steady state input impedance at the tuner output is 17 + j11 ohms.
I could not exactly match the tuner component configuration described by Jim
and get the correct input impedance.  This is not really relevant since I
arbitrarily set the component values such that the tuner's steady state
input impedance was 50 + j8 ohms.  I also set the components to be lossy
such that the final result for net power delivered to the transmission line
was somewhat consistent with Jim's experimental results.  I set QL = 150 and
QC = 800.  The component values I selected were series C at input = 115.71
pF; parallel L = 1.8 uH; and series C at output = 164.22 pF.  With these
tuner components, I calculated the steady state tuner input impedance to be
about 50 + j8 ohms.

I hope the following discussion helps describe the voltage analysis
presented in my Comm Quart articles.  Its important to note that all I know
about the steady state conditions in the analysis are the steady state load
impedance at the tuner output, the tuner component values and the fact that
the transmitter delivers 1000 watts of power to a 50 ohm load.

1)  The first point is to correctly determine the forward driving voltage
delivered to the transmission line.  Remember that when the "system" is
first energized no reflections are present at the tuner output.  Therefore,
the "load" impedance at the tuner output is Zo.  Therefore, the initial
"input impedance" at the tuner input is NOT 50 + j8 ohms, it is calculated
using circuit theory to be 58.1 - j51.7 ohms.     When the voltage of 223.6
volts arrives at the tuner input, a reflected voltage is created since an
impedance match does not exist.  The reflected voltage is equal to 55.2 -
j80.5 volts.  The reflected power is 190 watts.  Jim terminated the tuner
with a dummy load and measured the reflected power to be about 170 watts.

2)  The total voltage developed at the tuner input is equal to the sum of
the incident voltage and the reflected voltage.  This voltage is equal to
278.8 - j80.5 volts.  This is the driving forward voltage delivered to the
tuner input.  Using simple circuit theory, the driving forward voltage
delivered to the transmission line is determined to be -188.7 + j56.6 volts.
The driving forward power delivered to the transmission line is 776 watts.
This is the correct driving forward voltage and power delivered to the
transmission line.  Jim measured the power delivered to the dummy load
connected at the tuner output to be about 650 watts.  The measurement was
made with the 2500 watt plug.  It's possible that the meter reads low or my
assumption on tuner loss is low.

3) Now the wave reflection analysis begins.  I am not going to go into a lot
of detail here.  The driving forward voltage delivered to the transmission
line travels to the antenna.  A portion of it is reflected and arrives back
at the tuner output.  The level of voltage re-reflection at the tuner output
is entirely based upon the output impedance of the tuner.  This output
impedance is measured by terminating the input of the tuner with Zo (this is
based upon the assumption that the impedance seen looking towards the
transmitter from the tuner input is Zo) and then measuring the impedance
seen looking rearward into the tuner's output terminals.  In this example,
the tuner output impedance is determined to be 22 - j10 ohms.  The
reflection coefficient in this case (rho) is -0.36 - j0.18 with a magnitude
of 0.4.  Jim measured this rearward impedance to be about 17 - j7 ohms.
This is close.  The two factors affecting the measurements and the
calculation are the meter accuracy and the assumed tuner loss.  If the tuner
loss was ideally 0 dB, then this rearward impedance should measure and
calculate to be the conjugate of Zin = 17 + j11 (assuming a perfect match at
the tuner input).  As tuner loss increases, the measured and calculated
value will diverge from the conjugate of Zin.

4) After all of the wave reflections are considered, the level of reflected
voltage arriving at the tuner output is Vr = 90.5 - j89.3 volts which
translates to a reflected power of 323.2 watts, more or less consistent with
Jim's experiment.  The level of re-reflected voltage at the tuner output is
equal to rho*Vr = -48.8 + j15.3 volts.  The level of re-reflected power is
52.3 watts (NOT 323.2 watts). The total forward traveling voltage is equal
to the forward driving voltage (Vi) plus the re-reflected voltage (Vrer) and
is -237.5 + j71.9 volts.  The total forward traveling power is therefore
equal to 1231.4 watts, more or less consistent with Jim's experiment.  The
effective net power delivered to the transmission line is therefore 1231.4 -
323.3 = 908 watts, more or less consistent with Jim's experiment.  It is
important to note that the 776 watts of driving forward power and the 52.3
watts of re-reflected power are about .75 degrees out of phase (a perfect
match does not exist at the tuner input) and add to become the 1231.4 watts
of total forward power.

5) Last but not least.  What happens to the reflected power developed at the
tuner input?  Well, in the steady state, the total rearward traveling
voltage delivered back into the tuner is equal to the reflected plus
re-reflected voltage.  Using circuit theory, the resulting rearward
traveling voltage delivered back into the 50 ohm feeder transmission line
can be determined.  For a perfect steady state match to exist, this rearward
voltage would be the exact negative (180 degrees out of phase) with the
reflected voltage developed at the tuner input.  The two would cancel one
another and the net steady state rearward traveling power would go to zero.
Hence, no steady state reflected power.  In this analysis, the steady state
reflected power at the tuner output is calculated to be 6.4 watts, more or
less consistent with Jim's experiment.

The biggest issues from an analysis perspective is the assumed tuner loss.
I asked Jim to make some additional measurements to try and get a better
estimate of tuner loss but the meter reading are not accurate enough (see
Tom's - W8JI post).

Note that with this voltage and wave reflection analysis, the correct steady
state power levels are determined.  The correct relationships between
voltage and power are maintained and the wave reflections are always a
direct function of the impedance terminating the transmission line.

This message is more than long enough at this point.

73

Steve, VE9SRB


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