On May 25, 4:20 pm, walt wrote:
Hello Keith,

Thank you for your response. I’m starting my answer to your statements
by first quoting from one of your posts:

“Try as I might, I have not been able to derive a mechanism to explain
the observations in Reflections. But the explanations offered in
Reflections require large chunks of linear circuit theory to be
discarded, so this does not seem to be an appropriate path.”

That you have been unable to derive a mechanism that explains the
action in an RF power amplifier is evidence that you do not understand
it. So let’s examine the action that follows an appropriate path that
does not require any linear circuitry to be discarded. Further
evidence that you do not understand it is that you used a bench power
supply to describe the action, which you state has an infinite source
resistance when the load exceeds 50 ohm, and zero source resistance
when the load is less than 50 ohms. Unfortunately, this power supply
in no way resembles an RF power amplifier, either in components or
action.

“No way” is a bit strong. The RF PA is constructed from a constant
voltage source (the power supply) and a constant current controller
(the tube), both aspects present in the bench supply example
previously
offered. A tube is often modelled as an ideal variable constant
current
source but unlike an ideal source, which can produce whatever voltage
is needed to drive the current, the current produced by the tube is
limited by the power supply voltage. Thus, assertions of linear
behaviour need to be tempered by ensuring that such voltage limits
are not exceeded.

We’ll begin by stipulating that the ‘filter’ is a pi-network tank
circuit, having a tuning capacitor at the input and a loading-
adjustment capacitor at the output. We’ll also stipulate that the
plate voltage and the grid bias are set to provide the desired
conditions at the input of the tank circuit, which means that the
desired grid voltage is that which results in the desired conduction
time for the applied plate voltage. The result provides a dynamic
resistance RL, which is determined by the average plate voltage VPavg
and the average plate current IPavg appearing at the terminals leading
to the input of the tank circuit. In other words, RL = VPavg/IPavg.
To permit delivery of all available power to be delivered by the
dynamic resistance RL, we want the input impedance appearing at the
input of the tank circuit to be equal to RL.

Most references use Vpeak and Ipeak, though they are usually related
to average values with constants of proportionality so the computed
RL will be the same. None-the-less, I prefer peak values since it
ties better to the choices made in the design.

The power that can be controlled by a control device (be it a switch,
tube or transistor) is related to device limitations. So, for example,
the maximum power that can be controlled by a 250V 1A switch is 250W.
This occurs with a supply voltage of 250V and a load of 250 ohms.
Increasing the supply voltage exceeds the switch capability as does
reducing the load resistance.

If the supply voltage is less than 250V then the maximum power occurs
with a load that causes 1A to flow and is now a limit based on
circuit choices and device capabilities.

Note that these power limits have nothing to do with maximum power
transfer in a linear circuit.

Similarly in a tube circuit, the maximum power is limited by the
supply
voltage and the tube drive level (which sets the current that will
flow in the tube). Maximum controlled power is then V*I and occurs
with a load resistance of V/I. Increasing the load resistance reduces
the power because there is insufficient voltage from the supply to
drive more current through the load and reducing the load resistance
reduces the power because less voltage is impressed across the load.

Note that neither of these effects is related to the maximum power
transfer in a linear circuit.

We’ll now go to the output of the tank circuit. We’ll assume the load
to be the input of a transmission line on which there are reflections.
The result is that the input to the line contains a real component R
and a reactance jX. The output terminals of the tank circuit are the
two terminals of the output-loading capacitor. When the line is
connected to the output terminals of the tank circuit the reactance
appearing at the line input is reflected into the tank circuit. This
reactance is then cancelled by the tuning capacitor at the input of
the tank circuit, resulting in a resonant tank circuit. We now need to
adjust the output-loading capacitor to apply the correct voltage
across the input of the transmission line so that the real component R
appearing at the line input is reflected into the tank circuit such
that the resistance RL appears at the input of the tank circuit, thus
allowing all the available power to enter the tank circuit. In other
words, adjusting the loading capacitor to deliver all the available
power into the line also makes the output resistance of the tank
circuit equal to the real component R appearing at the line input.
With any other value of output resistance of the source, all the
available power would not be delivered to the line. A corollary to
that condition follows from the Maximum Power Transfer Theorem that
for a given output resistance of the source (the tank circuit), if the
load resistance is either increased for decreased from the value of
the source resistance, the delivery of power will decrease. This
condition also accurately describes the condition for the conjugate
match.

While a conjugate match does result in a situation where altering
the load will reduce the power transfer, it is not true that any
situation where altering the load reduces the power transfer is
also a conjugate match. The two examples above (bench power supply,
tube in a circuit) amply demonstrate this.

Keep in mind that the input impedance of the line is complex, or
reactive, but the reactance of the correctly-adjusted tuning capacitor
has introduced the correct amount of the opposite reactance to cancel
the reactance appearing at the line input. Thus the line input
impedance is R + jX and the output impedance of the source is R – jX,
providing the conjugate match.

This is quite in error, unless, by happenstance, RL is equal to Rp
(plus
the other contributors to source impedance).

You stated in one of your posts that the phase of the reflected wave
in relation to that of the source wave results in a non-linear
condition. This is totally untrue. The tuning action of the input
capacitor in the tank circuit that cancels the line reactance caused
by the reflection on the line in no way introduces any non-linearity
in the circuit, and the condition in the vicinity of the output of the
tank circuit is totally linear. Thus, circuit theorems that require
linearity to be valid are completely valid when used with the RF power
amplifier as described above. This applies to all RF power amplifiers,
Class A, AB, B and C.

For any circuit with a conduction angle of less than 360 degrees, my
simulations indicate otherwise. The reflection coefficient experienced
by the reflected wave when it arrives at the amplifier output varies
with the phase of the reflected wave. Since the reflection coefficient
is a function of source impedance and line impedance, and the line
impedance is not changing, this means that the source impedance is
changing with the phase of the reflected wave. This is not a behaviour
that is consistent with a linear circuit. Given the non-linearities
in a circuit with a conduction angle of less than 360 degrees, this
should not be a surprise. More, it would be a surprise if such a
circuit did behave as a linear circuit.

I hope my comments above assist in understanding the action that
occurs in RF power amplifiers.

Thank you. They have indeed helped clarify my thinking.

We are still left with the puzzle of why the observations documented
in Reflections report a reduction in power transfer when the load is
changed in either direction. It seems unlikely that RL is, by
happenstance, equal to Rp, which would be one explanation.

It seems plausible that it is related to the behaviours associated
with the examples I provided above, but I can not articulate a
mechanism that satisfies.

…Keith