[AMPS] Conjugate Matching In Class B and C Amplifiers
Tom Rauch
W8JI@contesting.com
Sat, 12 May 2001 00:31:56 -0400
> I hope I am in the ball park here, and have whetted a few appetites
> for more knowledge out there. Should Bruene choose to reply to
> Maxwell's latest article, I think we will find the main disagreement
> was how far back from the load the conjugate match occurs.
There can be no conjugate match in a non-linear system. Non-
linear in this context means a fractional cycle nonlinearity....not the
linearity we normally consider where the output power tracks the
input power.
One analogy of this is a water wheel driven by a limited source of
water pressure. At a certain load impedance (in the mechanical
case it is the ratio of shaft speed to torque) we are extracting
maximum energy from the water. The efficiency can be anything,
but maximum power transfer occurs when the speed of the falling
water and weight is matched by the load on the wheel. Try to
extract more torque, and the wheel slows down and horsepower is
reduced. Change things for more speed, and the torque falls off
along with horsepower.
A certain load impedance provides maximum energy transfer, and
it behaves like a lossless power limiting.
Now imagine a piston engine, with a finite chamber pressure and
volume driving pistons. There is no way to conjugately match that
source, because the pressure and speed are changing over
fractions of a cycle.
But once that pulsing is smoothed by a flywheel, it behaves similar
to the waterwheel. A certain optimum load impedance, or ratio of
torque to shaft speed, will extract maximum power from the
source. We have no idea what the efficiency is, only that we are
transferring maximum power.
PA's behave in a similar manner. I've measured it two ways.
One way was driving an operating PA backwards through a 30 dB
attenuator, and using a multiple tapped line over 1/4 wl long to
"view" the voltage of the reverse generator with a selective voltmeter
bridged across the line with a high impedance bridging pad. By
moving that pad from tap to tap, I could look at voltage along the
line towards the PA while the PA was running at full power. The
generator and PA were separated only enough in frequency for the
selectivity of the selective meter, a few hundred Hz.
When I adjusted the operating PA for maximum power transfer to
the load at a fixed amount of drive, voltage of the reverse generator
along the line going back to the PA was flat. This was true even
with a class C PA (I used a DX-100 Heathkit with extra grid bias
and some other mods to make the tubes switch hard, so efficiency
was very clearly well over 70%).
When drive was removed, the line was no longer flat. The reverse
generator's standing waves along the line looked terrible.
When I reduced drive, the conjugate match disappeared. It was
restored by peaking the pi-net for the new drive value.
More recently, at the request of Walt, I measured another rig. I
tested a T4XC. I used the load-pull method, where the load is
changed just the slightest amount and the power change is
measured. By measuring the voltage change across the load with a
bridge-type meter or any other very accurate meter, you can
calculate the source impedance driving the load when the load is
very slightly perturbed. I changed the load from 50 ohms to 49
ohms, and to 51 ohms.
When I used the standard formula for calculating source
"resistance" that PA looked like 50 ohms. When I mistuned the
PA, by overloading the PA, it looked like more than 50 ohms.
When I mistuned the PA by underloading it, it looked like less than
50 ohms. The results were similar to the reverse power test, except
with the reverse power test you can calculate the PA's impedance
by looking at the reverse generator's voltage distribution along the
tapped line.
Now certainly a PA does not need to be conjugately matched. But
maximum available power certainly occurs when it is...at least
according to what I have seen.
As a matter of fact if you look back at Bruene's original QST article
at the ETO amplifier, you will see it crosses 50 ohms at about
1200 watts. My bet is that is the power where he optimized the
tuning for maximum power transfer.
Now you can fuss with the knobs and make it look like almost
anything, but in every case I looked at maximum power transfer
occurred when the PA looked "flat" or nearly flat.
>From my old "Circuits and Networks" textbook, here's what it says:
"For many applications the purpose of inserting a network between
a source impedance Zg and load impedance ZL is to effect a
conjugate match. The purpose of this is, as is given in chapter 2
section 5, to deliver the optimum power to ZL. For this type of
match, the network is so designed that the impedance toward the
source at the output terminals 2a and 2b with the source voltage
zero is the conjugate of ZL."
Chapter 2 describes the Maximum Power Transfer Theorem. The
Maximum Power Transfer theorem states among other things
""Optimum power in ZL results when ZL is the conjugate of Zcd".
As I look through textbooks, it appears the first mention of
efficiency being limited occurs in the late 70's or early 80's. The
actual rules or descriptions of the theorems however clearly state
the theorems can not be used to define what happens inside the
source or at a point in the system that becomes non-linear. The
theorems describe source behavior as far as the load is concerned
for maximum power transfer, and clearly limit that to what the
terminals "look like" and not what is happen upstream where the
system might become non-linear.
It's kinda like moving from the wheel's loading by the road up
though the driveline of a car, and going past the flywheel. Once the
energy is pulsed and unsmooth, you can't define a fixed
impedance. At the point of the system where power is smoothed, it
might as well be from a perfectly linear source. We only know
where maximum energy transfer occurs, and not what the
efficiency of the conversion process is. Once past the flywheel
(tank circuit) everything is smooth, and the proportion of across
(voltage) and through (current) vectors is constant for a given tuning
condition.
We had the same thing with generators, where the field magnetism
was changed to effect a conjugate match between the generator
and load. If feedback was removed and the field held constant, one
optimum load impedance would extract maximum power from the
generator. While the purpose of the feedback loop through the
regulator was to hold the voltage constant, optimum generator
efficiency occurred when a load pull proved the generator
impedance matched the load impedance. The regulator placed the
generator close to that impedance by varying the field level,
although it could have been done with a fixed field level by varying
the shaft torque and speed.
There often a dozen ways to look at a problem, and the only flaw
happens when we take the model outside the limits of the model or
assume the model is the entire system. We take the model
literally when we should not. I think this is what happened with this
conjugate match stuff, and people who really haven't spent much
time researching the problem leap to conclusions by assuming the
model actually is the real world system.
They look at the generator in series with a resistance, and never
read the actual theorem behind the model. They then wrongly
conclude the model is reality, and calculate things inside the
model (like efficiency) when the theorem behind the model clearly
states it can NOT be used to describe anything happening inside
the terminals of the source.
Anyone who uses the Thevenin model to calculate efficiency limits
better go back and read the theorem, and the limits of the model. A
conjugate match means we are transferring optimum power when
the load looks like the complex conjugate of the source. Nothing
more, nothing less.
73, Tom W8JI
W8JI@contesting.com
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