On Wed, 28 Feb 2007 20:35:27 GMT, Owen Duffy wrote:

Breaking out of the previous thread to explore the "power explanation" in
a steady state situation:

The scenario for discussion is a transmitter connected to a half wave of
600 ohm lossless transmission line connected to an antenna with a
feedpoint impedance of 70+j0.

The transmitter is rated for 100W output, 100W is developed in the 70 ohm
load, the VSWR on the transmission line is 8.6, the "forward power"
(meaning Vf^2/Zo) on the transmission line is 267W, the "reflected
power" (meaning Vr^2/Zo) on the transmission line is 167W, the DC input
power to the transmitter is 200W.

The questions a

Is there any internal inconsistency in the scenario characterisation, if
so, identify / explain?

What is the heat dissipated in the transmitter (and why)?

What part of the "reflected power" of 167W is dissipated in the
transmitter (and why)?

Owen

Hi All,

Per recent correspondence from Walt Maxwell, he has asked me to post
his contribution:

Hi Richard,

I'm in a hotel in Jacksonville, away from my home computer, and at
this time I can't access the rraa to send, can only receive, so I'm
asking for your help.

I've been reading the posts on this thread and find it interesting.
However, it's been only discussed academically. On the other hand,
I've made measurements that prove the results described, measurements
made since those reported in Reflections 2.

I'd like for you to alert the posters on this thread to see Chapter
19A that will appear in Reflections 3, which is available for download
from my web page at www.w2du.com. The entire chapter was written as a
final epilogue to Bruene's fiction, but the portion pertinent to the
thread is in the last portion of the chapter concerning the
measurements made using a Kenwood TS-830S. Therein lies
the proof.

It would be nice if you could post the entire portion of the
measurements section, but that probably wouldn't work, because of
special characters used in Word that wouldn't appear in the text.
Anyway, I'd like for the posters to know that experimental proof
exists to support the claims made in the thread.

Thanks, Richard,

Walt


73's
Richard Clark, KB7QHC

Owen Duffy wrote:
"Breaking out of the previous thread to explore the "power explanation"
in a steady state situation:"

All OK as I see it.

Bird tells us that if you have significant standing waves, reflected
power is 10% or more of the forward power, and the ratio of reflected
power to forward power is then easily determined on the Bird Thruline
Wattmeter. Ratio of the reflected power to forward power is easily
converted to VSWR.

Bird supplies, charts, slide rules, and a formula for this conversion.

Bird confirms: "Power delivered to and dissipated in a load is given by:

Watts into load = Watts forward - Watts reflected."

Owen Duffy told us 100W is developed in 70 ohm load and the DC input
power of the transmitter is 200W.

Obviously 100W is dissipated in the transmitter and the efficiency is
50%.

Best regards, Richard Harrison, KB5WZI

On Wed, 7 Mar 2007 12:08:53 -0600, (Richard Harrison) wrote:

Owen Duffy wrote:
"Breaking out of the previous thread to explore the "power explanation"
in a steady state situation:"

All OK as I see it.

Bird tells us that if you have significant standing waves, reflected
power is 10% or more of the forward power, and the ratio of reflected
power to forward power is then easily determined on the Bird Thruline
Wattmeter. Ratio of the reflected power to forward power is easily
converted to VSWR.

Bird supplies, charts, slide rules, and a formula for this conversion.

Bird confirms: "Power delivered to and dissipated in a load is given by:

Watts into load = Watts forward - Watts reflected."

Owen Duffy told us 100W is developed in 70 ohm load and the DC input
power of the transmitter is 200W.

Obviously 100W is dissipated in the transmitter and the efficiency is
50%.

Best regards, Richard Harrison, KB5WZI

Much has been said in this thread concerning whether reflected power does or does not enter the power
amplifier and cause heating of the plate. In describing below the experimental procedure that I have performed
many times in my RF Lab, I am offering proof here that such heating does not occur.

The following material from Chapter 19A of Reflections 3 is presented to show why reflected power
does not enter the amplifier and cause heating of the tube plate.

"We’ll now examine the experimental data that resulted from measurements performed
subsequent to those reported in Chapter 19, new data that provides additional evidence that a
conjugate match exists at the output terminals of an RF power amplifier when all of its
available power is delivered into its load, however complex the load impedance. According to
the definition of the conjugate match as explained earlier, if this condition prevails there is a
conjugate match. In addition, the data presented below also provides further evidence that the
output source resistance of the RF amplifier is non-dissipative. CONSEQUENTLY, THIS EVIDENCE
PROVES THAT REFLECTED POWER DOES NOT CAUSE HEATING OF THE PLATE. The following steps describe
the experimental procedure I employed and the results obtained:

1. Using a Kenwood TS-830S transceiver as the RF source, the tuning and loading of the pi-
network are adjusted to deliver all the available power into a 50 + j0-ohm load with the grid
drive adjusted to deliver the maximum of 100 watts at 4 MHz, thus establishing the area of
the RF power window at the input of the pi-network, resistance RLP at the plate, and the
slope of the load line. The output source resistance of the amplifier in this condition will
later be shown to be 50 ohms. In this condition the DC plate voltage is 800 v and plate
current is 260 ma. DC input power is therefore 800 v x 0.26 a = 208 w. Readings on the
Bird 43 wattmeter indicate 100 watts forward and zero watts reflected. (100 watts is the
maximum RF output power available at this drive level.) From here on the grid drive is left
undisturbed, and the pi-network controls are left undisturbed until Step 10.

2. The amplifier is now powered down and the load resistance RL is measured across the input
terminals of the resonant pi-network tank circuit (from plate to ground) with an HP-4815
Vector Impedance Meter. The resistance is found to be approximately 1400 ohms. Because
the amplifier was adjusted to deliver the maximum available power of 100 watts prior to the
resistance measurement, the average of the dynamic resistance RLP looking into the plate
(upstream from the network terminals) is also approximately 1400 ohms. Accordingly, a
non-reactive 1400-ohm resistor is now connected across the input terminals of the pi-network
tank circuit and output source resistance Ros is measured looking rearward into the output
terminals of the network. Resistance Ros was found to be 50 ohms.

3. Three 50-ohm dummy loads (a 1500w Bird and two Heathkit Cantennas) are now connected
in parallel to provide a purely resistive load of 16.67 ohms, and used to terminate a coax of
13.5° length at 4 MHz.

4. The impedance ZIN appearing at the input of the 13.5° length of coax at 4 MHz terminated
by the 16.67-ohm resistor of Step 3 is measured with the Vector Impedance Meter, and
found to be 20 ohms at +26°. Converting from polar to rectangular notation, Zin = 17.98 +
j8.77 ohms. (Zin = Zload from the earlier paragraphs.) This impedance is used in Steps 5
and 6 to provide the alternate load impedance in the load-variation method for determining
the complex output impedance of the amplifier, and for proving that the conjugate match
exists.

5. With respect to 50 ohms, Zin from Step 4 yields a 2.88:1 mismatch and a voltage reflection
coefficient (rho) 0.484. Therefore, power reflection coefficient (rho squared) 0.235, transmission
coefficient (1 - 0.235) = 0.766, and forward power increase factor 1/(1- 0.235) = 1/0.766 = 1.306.

6. Leaving pi-network and drive level adjustments undisturbed, the 50-ohm load is now
replaced with the coax terminated with the 16.67-ohm load from Step 4, thus changing the
load impedance from 50 ohms to 17.98 + j8.77 ohms, the input impedance Zin of the coax.

7. Due to the 2.88:1 mismatch at the load, neglecting network losses and the small change in
plate current resulting from the mismatch, approximately the same mismatch appears
between RLP and ZL at the input of the pi-network. Consequently, the change in load
impedance changed the network input resistance RL from 1400 ohms to complex ZL = 800 -
j1000 ohms, measured with the Vector Impedance Meter using the method described in Step
2. To verify the impedance measurement of ZL the phase delay of the network was
measured using an HP-8405 Vector Voltmeter and found to be 127°. Using this value of
phase delay the input impedance ZL was calculated using two different methods; one
yielding 792 - j1003 ohms, the other yielding 794.6 - j961.3 ohms, thus verifying the
accuracy of the measurement. However, because grid voltage Ec, grid drive Eg, and plate
voltage Eb are left unchanged, the average dynamic resistance RLP at the plate has remained
at approximately 1400 ohms, leaving a mismatch between RLP and ZL at the input of the pi-network.
As stated above, this value of ZL yields the substantially the same mismatch to plate resistance
RLP as that between the output impedance of the pi-network and the 17.98 + j8.77-ohm load, i.e.,
2.88:1. This mismatch at the network input results in less power delivered into the network,
and thus to the load, a decrease in the area of the RF window at the network input, and a
change in the slope of the loadline. (It must be remembered that the input and output
mismatches contribute only to mismatch loss, which does not result in power delivered and
then lost somewhere in dissipation. As we will see in Step 8, the mismatch at the input of
the pi-network results only in a reduced delivery of source power proportional to the
degree of mismatch.)

8. Readings on a Bird 43 power meter now indicate 95w forward and 20w reflected, meaning
only 75 watts are now delivered by the source and absorbed in the mismatched load. The
20w reflected power remains in the coax, and adds to the 75 watts delivered by the source to
establish the total forward power of 95w.

9. We now compare the measured power delivered with the calculated power, using the power
transmission coefficient, (1 - 0.235). The calculated power delivered is: 100w x (1 - 0.235) = 76.6w,
compared to the 75w indicated by the Bird wattmeter. However, because the new load
impedance is less than the original 50 ohms, and also reactive, the amplifier is now
overloaded and the pi-network is detuned from resonance. Consequently, the plate current
has increased from 260 to 290 ma, plate voltage has dropped to 760 v, and DC input power
has increased from 208 w to 220.4 w.

10. With the 17.98 + j8.77-ohm load still connected, the pi-network loading and tuning are
now re-adjusted to again deliver all available power with drive level setting still left
undisturbed. The readjustment of the plate tuning capacitor has increased the capacitive
reactance in the pi-network by -8.77 ohms, canceling the +8.77 ohms of inductive reactance
in the load, returning the system to resonance. The readjustment of the loading control
capacitor has decreased the output capacitive reactance, thus reducing the output resistance
from 50 to 17.98 ohms. Thus the network readjustments have decreased the output
impedance from 50 + j0 to 17.98 - j 8.77 ohms, the conjugate of the load impedance,
17.98 + j8.77 ohms. The readjustments have also returned the network input impedance ZL
to 1400 + j0 ohms (again equal to RLP), have returned the original area of the RF window at
the network input, and have returned the slope of the loadline to its original value. For
verification of the 1400-ohm network input resistance after the readjustment, ZL was again
measured using the method described in Step 2, and found it to have returned to 1400 + j0
ohms.

11. Bird 43 power meter readings following the readjustment procedure now indicate 130w
forward and 29.5w reflected, indicating 100.5w delivered to the mismatched load.

12. For comparison, the calculated power values a Forward power = 100 x 1.306 = 130.6w,
reflected power = 30.6w, and delivered power = 130.6w - 30.6w = 100w showing
substantial agreement with the measured values. (1.306 is the forward power increase factor
determined in Step 5.) Plate current has returned to its original value, 260 ma, and likewise,
plate voltage has also returned to the original value, 800 v. Consequently, the DC input
power has also returned to its original value, 208 w.

13. It is thus evident that the amplifier has returned to delivering the original power, 100 watts
into the previously mismatched complex-impedance load, now conjugately matched, the
same as when it was delivering 100 watts into the 50-ohm non-reactive load. But the
reflected power, 30.6 watts, remains in the coax, adding to the 100 watts delivered by the
amplifier to establish the 130.6 watts of forward power, proving that it does not enter the
amplifier to dissipate and heat the network or the tube.

It must be kept in mind that impedance Zin appearing at the input of the 13° line connecting
the 16.7-ohm termination to the output of the amplifier is the result of reflected waves of both
voltage and current, and thus reflected power is returning to the input of the line, and becomes
incident on the output of the amplifier.
The significance of these measurement data is that for the amplifier to deliver all of its
available power (100w) into the mismatched load impedance Zin = 17.98 + j8.77 ohms, the
readjustment of the tuning and loading of the pi-network simply changed the output
impedance of the network from 50 + j0 ohms to 17.98 - j8.77 ohms, the conjugate of the
load impedance, thus matching the output impedance of the network to the input impedance
of the coax. Consequently, there IS a conjugate match between the output of the transceiver
and its complex load. QED. The readjustments of the pi-network simply changed its
impedance transformation ratio from 50:1400 to (17.98 - j8.77):1400, returning the input
resistance RL of the pi-network to 1400 ohms, the value of RLP. Thus the plates of the amplifier
tubes are unaware of the change in external load impedance.

14. We’ll now make an additional indirect measurement of Ros that proves the conjugate
match statement above is true. Leaving the pi-network adjustments undisturbed from the
conditions in Step 10, with the amplifier powered down we again connect a 1400-ohm non-
reactive resistor across the input terminals of the pi-network tank circuit and measure the output source
impedance Zos looking rearward into the output terminals of the network. The impedance was
found to be Zos = 18 - j8 ohms.

From a practical viewpoint, measured impedance Zos = 18 - j8 ohms is the conjugate of
load impedance Zload = 17.98 + j8.77, proving that the amplifier is conjugately matched to
the load, and also proving the validity of the indirect method in determining that the source
impedance of the amplifier is the conjugate of the load impedance when all available power is
being delivered to the load.
Thus the data obtained in performing Steps 1 through 14 above proves the following four
conditions to be true:

No reflected power incident on the output of the amplifier is absorbed or dissipated in
the amplifier, because:

1. The total DC input power is the same whether the amplifier is loaded to match the resistive
Z0 load of 50 + j0 ohms, with no reflected power, or to match the complex load of 17.98 -
j8.77 ohms with 30.6 watts of reflected power, while 100 w is delivered to either the Z0 load or
the re-matched complex load.

2. All the 100 watts of power delivered by the transmitter is absorbed in both the Z0 load and
the re-matched complex load cases, with the same DC input power in both cases.

3. All the 30 watts of reflected power has been shown to add to the source power, establishing
the total 130 watts of forward power in the case involving the re-matched complex load.

4. All the reflected power is added to the source power by re-reflection from the non-
dissipative output source resistance Ros of the amplifier. Had the output source resistance of
the amplifier been dissipative the reflected power would have been dissipated there to heat,
instead of being re-reflected back into the line and adding to the source power. In addition, the
Bird 43 power meter would have indicated 75 watts of forward power, not 95. This proves that
reflected power incident on the output of the amplifier does not cause heating of the tube.

It should also be noted, an accepted alternative to the load-variation method for measuring
the output impedance of a source of RF power is the indirect method demonstrated above. As
performed during the measurements described above, the procedure for this method is to first
make the necessary loading adjustments of the output network to ensure that all of the
available power is being delivered to the load. Next, the input impedance of the load is
measured. It then follows that, as proven above, the source impedance is the conjugate of the
input impedance measured at the input of the load, because when all available power is being
delivered to the load, this condition conforms to the Conjugate Matching and the Maximum
Power-transfer Theorems.
Additionally, I previously performed this same measurement procedure using a HeathKit
HW-100 transceiver, using several different lengths of coax between the 16.7-ohm load and
the output of the transceiver in each of several measurements. The different lengths of coax
provided different complex load impedances for the transceiver during each measurement. The
same performance as described above resulted with each different load impedance, providing
further evidence that a conjugate match exists when the amplifier is delivering all of its
available power into its load. These results also prove that the single test with the Kenwood
transceiver is not simply a coincidence.

More recent experimental evidence has been presented since that of Chapter 19, adding
further proof that a conjugate match can exist when the source is an RF power amplifier, and
that the output source resistance of the amplifier is non-dissipative. It was also shown that
the average dynamic resistance RLP looking toward the plate from the network input equals
resistance RL appearing at the input of the pi-network when a conjugate match is obtained,
while contrary to Bruene’s claim, there is no requirement that RL = RS to obtain a conjugate match,
thus proving Bruene’s definition of the conjugate match appearing in his November 1991 QST
article 142 invalid.

Walt, W2DU

I wrote:
"Obviously 100W is dissipated inthe transmitter and efficiency is 50%."

This is a Class A amplifier limit but not for other classes of
amplifiers. Terman tells us on page 450 of his 1955 opus:
"The high efficiency of the Class C amplifier is a result of the fact
that plate current is not allowed to flow except when the instantaneous
voltage drop across the tube is low; i.e., Eb supplies energy to the
amplifier only when the largest portion of this energy will be absorbed
by the tuned circuit.

"Transmission Lines, Antennas, and Wave Guides" by King, Mimno, and Wing
is an excellent reference, and like Terman, the authors agree with
Walter Maxwell. On page 43 is found:
"Principal of Conjugates in Impedance Matching - If a dissipationless
network is inserted between a constant-voltage generator of impedance Zg
and a load of impedance ZR such that maximum power is delivered to the
load, at every pair of terminals the impedance looking in opposite
directions are conjugates of each other."

The real world is full of imperfections which by no means preclude
practical work.

Best regards, Richard Harrison, KB5WZI

Richard Harrison wrote:

"Transmission Lines, Antennas, and Wave Guides" by King, Mimno, and Wing
is an excellent reference, and like Terman, the authors agree with
Walter Maxwell. On page 43 is found:
"Principal of Conjugates in Impedance Matching - If a dissipationless
network is inserted between a constant-voltage generator of impedance Zg
and a load of impedance ZR such that maximum power is delivered to the
load, at every pair of terminals the impedance looking in opposite
directions are conjugates of each other."

And can be seen clearly by looking at the reflection of a smith chart
upside down.

Best, Dan.