Owen,

Just returned from five days in Jacksonville, FL for spinal surgery that didn't
occur. I had cataract surgery on Feb 20, and the spinal surgeon said I needed
six weeks of recovery from the earlier surgery before he'd perform the spinal,
for fear of damaging the yet unhealed eye. So now I can get back to business as
usual for a while.

On Wed, 07 Mar 2007 23:22:28 GMT, Owen Duffy wrote:

"walt" wrote in
oups.com:

Walt,
...
First, let me say that although the average source resistance at the
plates appears to be 1400 ohms in the case I described, and IMHO I
believe it is, I'm not in the position of stating that is as a fact.

Ok, I think we are agreed that the measurements haven't directly
supported that belief.

What I do claim as a fact is that when the transmitter is loaded to
deliver all available power to its load, the OUTPUT source resistance
(or impedance) at the output terminals is the conjugate of its load.

If it were a linear source and you delivered *maximum* (as opposed to
*all*) to the load, I agree that the load impedance is the complex
conjugate of the source impedance. That is essentially the Jacobi Maximum
Power Transfer Theoram.

I'm familiar with the Maximum Power Transfer Theorem that appears in Everitt's
'Communication Engineering', but not with Jacobi's. Is there a difference in
their definitions?

I still believe that the 1400 ohms appearing in my paper you reviewed is the
average of the dynamic resistance of the source, because the maximum power
available for the given drive level, i.e., the 'available' power, was being
delivered when the input resistance of the pi-network was 1400 ohms, while less
than the available power was delivered when the input resistance of the network
was either greater or less than 1400 ohms. This condition conforms to the
Maximum Power Transfer Theorem as I understand it.

The question is whether it is a sufficiently linear source to use that
model.

I'm differentiating between the conditions at the input of the pi-
network and those at the output, because the energy storage effect of
the network Q isolates the output from the input, such that the
conditions at the output can be represented by an equivalent Thevenin
generator. At the output terminals the conditions appearing at the
input are irrelevant, such as the shape and duration of the voltage
applied to the pi-network, as long as the energy storage Q is
sufficient to support a constant voltage-current relationship (linear)
at the output for whatever load is absorbing all the available power
from the network.

Thus, when all available power is delivered into a 50-ohm load the
source resistance at the output terminals is 50 ohms. Please also
review the later portion of Chapter 19, also available on my web page.
On those pages I report the results of measurements using the load-
variation method, which also shows the output source resistance to
equal the load resistance when the amp is delivering all its available
power.

Walt, I have just re-read that section and note your measurements which
explored the delta V and delta I for small load variation (delta R) where
delta R is always negative, and calculated results.

Your results are interesting.

I have seen others report quite different results, and have found
differently myself on rough measurements, but I note your comments on the
sensitivity of the calculated Rs to tuning/matching which might reveal
why other tests disagree.

Owen, you wouldn't believe the sensitivity of these measurements unless you
tried them.

First, measuring voltages v1 and v2 with an analog voltmeter just won't cut it.
A VM with digital readout is essential, and it must be precise to 0.1 v. Because
the delta current in the denominator is so small, even an error of 0.2 v skews
the result.

Second, reading the voltage at each load requires a few seconds for the voltage
and current to stabilize, and then the readings must be taken while the voltage
and current are stabile for at least two to three seconds. You'd be surprised
how unstable the voltage is at the 0.1 v measurement level. Even so, many
readings must be taken, first the reference, then the load, then average out the
readings.

Third, consider the breadth of the slope of the power vs load resistance curve
at the peak power point--it is broad! For example, a load resistance of +/- 10%
of the source resistance, say 55 or 45 ohms, yields a 1.11111 mismatch to the
source, for a reflection coefficient of 0.053, for a power loss of only 0.012
dB. When trying to adjust the amp loading for maximum delivery of power using an
analog meter, it's impossible to observe a change of 0.01 dB, thus when one
appears to have loaded for maximum, the source resistance can be anywhere
between 45 and 55 ohms and you'd never know just what it is. Thus, it's
practically impossible to achieve a perfect conjugate match in practice, but you
could. On the other hand, though not a perfect conjugate match, you'll have a
practical conjugate match, because the difference in power delivered is
insignificant. What I'm trying to convey is that a conjugate match is possible,
though difficult to achieve when loading the xmtr.

Consequently, I can appreciate why others have obtained results different from
mine, unless they have taken the necessary steps to overcome the sensitivity
problem of the small current number in the denominator of the load-variation
equation.

(It only takes one sound repeatable experiment that shows that the source
impedance is not the conjugate of the load to disprove the generality.)

On a practical note, the sensitivity discussed above does mean that if
your assertion about matching is true, it is unlikely that transmitters
are exactly matched.

See my above comment.

My measurements have been on transistor PAs with broadband transformer
coupling to the load. The transmitters have had a lowpass filter with a
break point well above operating frequency between the transistors and
load. It is a different configuration, and although my measurements were
rough, they indicated different apparent source impedance at different
drive levels which questions the linear model for large signal operation,
especially for modes with varying amplitude such as SSB telephony.

Owen

I should have mentioned above that all of my measurements that determined a
particular source resistance were taken with constant drive level, If
source-resistance measurements are taken at any one drive level, the observed
source resistance will be equal to the load resistance if all available power
was delivered during the measurements. But if the drive level is changed without
readjusting the loading, the source resistance will also have changed. However,
I have found that for any given drive level, the measured source resistance will
equal the load resistance if the loading is adjusted so that all the power
available for that drive level is being delivered.

Just to be sure we're on the same page, let me define my understanding of
'maximum' power and 'available' power. For any given drive level there is a
maximum of power that can be delivered. I call that the 'available' power. I
consider 'maximum' power to be that which can be obtained by overloading a tube
with excessive voltage or current relative to the manufacturer's ratings.

Walt, W2DU