Keith,

Assertions have been made about the Thevenin equivalent source impedance
(Zs) of an RF PA. In the discussions over the years, some people have
imposed some qualifications (eg classes, tuned / untuned etc).

Zs can be measured by a number of methods. It is interesting to note that
in many of the discussions on the 'net on this topic, more focus is given
to dismissing the experiment design for experiments that produce
unfavourable results to some, than questioning the proposition that Zs is
50+j0 or thereabouts.

Whilst Walt has documented a quite rigourous experiment, and it produced
a favourable outcome, it is my view that that in itself is not proof of
the proposition.

Whilst one favourable experiment cannot proove the proposition correct,
just one valid experiment can prove the proposition to be not generally
true. No doubt the reason for the focus on proving experiments invalid.

If you give some thought to what you could use Zs for, then every valid
experiment that is designed around that application must produce a
favourable outcome if Zs is as proposed. If they don't, then the value
assumed for Zs must be wrong.

I posted a simple test earlier that in my experience does not support the
proposition that typical ham HF transmitters have Zs=50+j0 or close to
it. The test is documented so that individuals can try it and make their
own mind up. Some might get favourable results on limited trials... but
in my view, that is outweighed by unfavourable results from valid
experiments.

Owen

Owen, I have reviewed your simple test using an ATU to obtain various
load impedances, but I don't see any data showing the results of any
tests you may have made using this procedure. I'm assuming it's the
same procedure I used in measuring the data I presented in Chapter 19
of Reflections 2, in which I varied the load impedance by switching
between two different, but closely related values of load resistance.

Unless we're discussing RF power amps using solid-state components, I
don't understand why anyone would expect the source impedance to be 50
+ j0 ohms unless the source was delivering all its available power
into a 50 + j0 load.

I've made an addition to Chapter 19 as it appears in Reflections 2.
That addition now appears in the new edition, Reflections 3. In that
addition I report the procedure and results of measuring the output
impedance of a Kenwood TS-830S terminated in a complex load of 17.98 +
j8.77 ohms. I described fourteen steps in the procedure that resulted
in the measured output impedance of the Kenwood to be 18 - j8 ohms,
which is sufficiently close to the conjugate of the load to be a
practical conjugate match between the source and load impedances.

Why then do you discount the data from my measurements as not proof
that the 18 - j8 ohms of source impedance is valid?

I'm going to try to insert a copy of the 14 steps of the procedure.
However, as I have tried this type of insertion previously and failed,
it also may fail here. If it fails I invite you to go to my web page
at www.w2du.com and click on the line "Preview Chapters from
Reflections 3" and then click on Chapter 19A, then scroll down to Sec
19A.5 Additional Experimental Data. There you will see the detailed
description of the entire procedure. I hope this will help in
understanding my posistion concerning the measurement of the source
impedance at the output of the tank circuit of an RF power amplifier.

Sec 19.14 Additional Experimental Data

The source resistance data reported in Secs 19.8 and 19.9 were
obtained using the load variation method with resistive loads. Note
that of the six measurements of output source resistance reported in
Table 19.1, the average value of the resistance is 50.3 ohms obtained
with the reference load resistance of 51.2 ohms, exhibiting an error
of only 1.8 percent. However, various critics assert that proof of a
conjugate match between the source and load requires the load to
contain reactance. Accordingly, the experimental data reported below
were obtained using both the load variation method and an indirect
method for determining the source impedance of the RF power amplifier,
with a resistive load to obtain a reference source resistance and a
complex load to determine the complex source impedance that is then
proven to be the conjugate of the complex load.
We’ll now examine the experimental data that resulted from
measurements performed subsequent to those reported in Secs 19.8 and
19.9, 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 in Sec 19.1, Axioms 1 and 2, 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. 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  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 averaged 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 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^2 = 0.235, transmission coefficient (1 –
rho^2) = 0.766, and forward power increase factor 1/(1 – rho^2) =
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 + j0 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, although grid
voltage EC, grid drive EG, are left unchanged, resistance RLP of
approximately 1400 ohms at the plate has changed somewhat due to the
small changes in plate voltage and plate current due to the change in
the load, 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 – rho^2. The
calculated power delivered is: 100w x (1 – rho^2) = 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 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 into
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
Theorems17.
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.

Walt, W2DU