[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 

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

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