[TowerTalk] Walt Maxwell responds to Steve Best
Steven Best
sbest@cushcraft.com
Thu, 11 May 2000 23:26:55 -0400
Hi Jim:
Thanks for posting Walt's note. I will respond as well. Hopefully we can
all keep this discussion focused on the technical points. As Walt did, I
will insert my comments within the text.
W2DU:
The impedance one measures looking
back into the matching device is entirely different from
the impedance seen by waves reflected from a
mismatched line termination on reaching the matching
device when the forward waves from the source are
present simultaneously. This is one of the many points
Steve fails to understand, even after I've tried to explain
to him with no success.
Steve's Response:
The impedance a traveling wave sees at the end of a transmission line does
not change because of the presence of source voltage. Walt's perception
here is similar to his perception of line impedance and power determination
discussed below. Remember that from the initial to the steady state there
are numerous wave reflections in the transmission line. Does the impedance
of the antenna change for a forward wave arriving at the antenna when a
reflected wave is already present at the antenna? NO!! This is a
conceptual issue that requires a lot more time than I can afford in this
e-mail. The other comments below are easier to address.
W2DU:
Steve's error here is in using the characteristic impedance
Zo to apply the voltage. Here Zo is irrelevant, because
when a 50-ohm line has reflections there is no place on
the line where the line impedance is 50 + j0 ohms. There
are two locations on the line where the line impedances
are purely resistive, 150 + j0 and 16.667 + j0 ohms. There
are also two locations on the line where the resistive
component of the line impedance is 50 ohms, but they
are complex line impedances, 50 + j57.7 and 50 - j57.7ohms.
Steve does not use the line impedance in his calculations
-he uses Zo, the CHARACTERISTIC impedance of the
line, which is incorrect. The result of this error? His
calculations using voltage with Zo results in a totally
erroneous value of forward power, 248.8 watts instead
of the correct 133.33 watts.
Steve's Response:
First, I would like to remind everyone that I agree that the correct steady
state forward power is 133.3 watts. My issue is that it is not a result of
100 watts adding in-phase to 33.33 watts. If you want, forget about Zo and
use any impedance Walt wants to use. Calculate the voltage magnitude
associated with the 100 watts. Calculate the voltage magnitude associated
with the 33.33 watts. Add the voltages together in phase and see what the
resulting power is.
Walt is quite correct in stating that there is no place along the length of
the transmission line where the line impedance is 50 + j0 ohms. Line
impedance is the steady state result of how the forward and reflected
voltages and currents develop at any point in the transmission line. Walt
says that my use of Zo is incorrect in determining forward or reflected
power.
Here's how it works. (This material can be found in any book on
transmission lines.)
At any point along the length of a transmission the ratio Vforward/Iforward
= Zo and the ratio Vreflected/Ireflected = Zo. Assuming Zo is real, the
forward traveling power is |Vforward|^2/Zo and the rearward traveling power
is |Vreflected|^2/Zo. The net forward power is Pforward - Preflected.
Line impedance at any point along the line is simply the ratio of total
voltage to total current at that point. Vtotal = Vforward + Vreflected and
Itotal = Iforward - Ireflected.
W2DU:
Steve's calculation of 248.8 watts of
forward power is incorrect, because of using Zo as the
line impedance, which it is not, instead of the line
impedance determined by the position of the standing
waves on the line.
Steve's Response.
I agree that the 248.8 watts is not the correct steady state forward power.
It is however, the power you get when you add 100 watts in phase with 33.33
watts. Let's assume that the transmission line is 1 wavelength long and
lossless. The line impedance would be 150 ohms. P=|V|^2/Z. With 100
watts, |V| = 122.47 v. With 33.33 watts, |V| = 70.71 v. Add these voltages
in phase (assume both are 0 phase) the total voltage is 193.18. Ptotal =
248.8 watts.
In a lossless transmission line with a mismatched antenna connected at the
end, the line impedance changes throughout the length of the entire line.
In a lossless transmission line, the magnitude of the forward traveling
voltage and current never change from the line input to the line end. The
forward traveling power never changes from the line input to the line end.
Now, if you were to use the line impedance to calculate the forward
traveling power from the forward traveling voltage it would not remain
constant throughout the line. Realize that power is also given by |V| |I|
cos(theta), where theta is the phase angle of the impedance. In the line,
the phase angle of the line impedance changes as you move along the line.
Calculating forward or reflected traveling power using forward or reflected
traveling voltage with line impedance does not work.
You can however, calculate the net power (Pforward - Preflected) at any
point along the line with the line impedance. In this case, net power is
given by Pnet = |Vtotal| |Itotal| cos(theta), where theta is the phase angle
of the line impedance. In the example we have been discussing, this
calculation would yield a net power of 100 watts at any point along the
line.
W2DU:
I will now explain why 133.33 watts
is correct, using for a reference the work of Princeton
University's Walter Johnson's "Transmission Lines
and Networks."
There isn't
sufficient space here to explain that concept, ...
Steve's Response:
I don't have any problem with 133.33 watts being the correct forward power.
I also don't have any problem with Johnson's text. It's Walt's
interpretation of Johnson's material where I have the problem. And yes,
there isn't sufficient time here to explain the concepts.
W2DU:
Steve said: "Two "in-phase" powers (P1 and P2) never add to become
the algebraic sum of P1 + P2. "
The above statement is totally untrue. When the
corresponding voltages and currents of the forward and
reflected powers are in phase the two powers add
algebraically. In the paragraph below Steve talks about
powers where the voltage and currents are out of phase.
This is irrelevant, because when the antenna tuner is
correctly adjusted to deliver maximum output power the
voltage and current phases are exactly equal at zero degrees.
Steve Response:
How many times do we need to do this calculation? Take two voltages of zero
phase. Calculate the power for each into any impedance. Now add the two
voltages together in phase as they would add in a transmission line. Now
calculate the total power. Is it the sum of P1 + P2? NO!! Let's do it
again. 100 volts is delivered to a 50 ohm load. P = 200 watts. 50 volts
is delivered to a 50 ohm load. P = 50 watts. Add both voltages. 150 volts
is delivered to a 50 ohm load. P = 450 watts. Power is a function of
voltage squared. It should be obvious that (V1 + V2)^2 is greater than V1^2
+ V2^2.
W2DU:
Sorry, it's Steve's concept and math that fall apart, not mine,
for the reason explained above, where he gets 248.8 watts
of forward power. There is no way there can be 248.8 watts
of forward power. See my explanation above showing that
he used the wrong value of Z to apply the voltage in
attempting to calculate the forward power. It's Steve's
relationships between voltage and power that are not
consistent with transmission line or circuit theory.
Steve's Response:
I agree that the forward power is not 248.8 watts. I know that it's 133.33
watts. Walt is absolutely correct that there cannot not be 248.8 watts of
forward power and for that reason alone we know that the forward power
cannot be determined by adding 100 watts and 33.33 watts.
W2DU:
It is in this article that he
totally and inconceivably trashed the concepts of reflection
mechanics I discussed in Reflections. And reviewing the
last sentence in his paragraph, can you really believe
that 75 watts plus 8.33 watts yields 133.33 watts. Where
did he learn that kind of arithmetic?
Steve's Response:
75 watts + 8.33 watts = 133.33 watts. This is getting fun. Let's all say
this together - power is a function of voltage squared. 75 watts into 50
ohms (pick any Z) is a voltage magnitude of 61.24 volts. 8.33 watts into 50
ohms is a voltage magnitude of 20.41 volts. The forward traveling voltages
add in phase to become 81.65 volts. The resulting total forward power is
... 133.33 watts!
W2DU:
Nearly all of the following of Steve's statements are untrue
and misleading, and prove that Steve has a large
misunderstanding of transmission line theory to overcome.
Steve's Response:
I understand that forward traveling voltages in a transmission line add. I
also understand that power is a function of voltage squared.
73,
Steve, VE9SRB
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