The Aikido Comedy (pt 2)

To take John Broskie's original 2004 post in hand,
where the "Aikido Amplifier" name was coined,
we can see John's own understanding of how the circuit operates.
We will only quote for review what is necessary for understanding the concepts:

Quote:

New Tube Circuit: The Aikido Amplifier "...this amplifier sidesteps power supply noise by incorporating the noise into its normal operation. As a result, in terms of distortion and output impedance and PSRR, the following circuit works at least a magnitude better than the equivalent SRPP or grounded-cathode amplifier. The improved PSRR advantage is important, for it greatly unburdens the power-supply design and it helps prevent the signal from recirculating through the power supply." ... How it works This circuit eliminates power-supply noise from the output, by injecting the same amount of PS noise at the top and bottom of the two-tube cathode follower circuit. The way it works is that the input stage (the first two triodes) define a voltage divider of 50%, so that 50% of the PS noise is presented to the CF's grid; at the same time the 100k resistors also define a voltage divider of 50%, so the bottom triode's grid also sees 50% of the PS noise. Since both of these signals are equal in amplitude and phase, they cancel each other out, as each triodes sees an identical increase in plate current (imagine two equally strong men in a tug of war contest). If the output connection is taken from the the cathode follower's cathode, then the balance will be broken. The same holds true if the cathode follower's cathode resistor is removed. (Besides, this resistor actually makes for a better sounding cathode follower, as it linearizes the cathode follower at the expense of a higher output impedance.) Note also the absence of any cathode resistor bypass capacitors; these caps are very much in the signal path and very few do not damage the sound, unless high quality capacitors are used. If a cathode resistor bypass capacitor is used on the input stage's bottom triode, then the two resistor voltage divider ratio must be changed from 50% to match the new AC noise divider ratio imposed by the input stage. In other words, much less PS noise noise will need to be injected into the cathode follower's bottom triode's grid. "

Certainly John's description is virtually enclosed, and self-explanatory.
Nor has he felt it necessary to edit his post in nearly 8 years.

 John's banter is so reasonable and pleasant,
its hard to imagine there could be any fundamental flaw
in his reasoning process.

But we are compelled to look closer at analysis.

The first thing we must ask, is,
What really is happening here?

(1) Is John giving a real analysis, or more of an 'analogy'?

(2) Is John knowingly oversimplifying the explanation?

I think the answer in 2004, by Mr. Broskie's own admission,
is that his understanding of this circuit was naive,
but intuitively 'lucky'.
Also, as the story unfolds, John reveals that he himself was surprised,
at the glowing reports given of the performance of a few of his designs.
Thus his interest and explanation may have been in part ad hoc,
a quick attempt to explain for himself and others why the circuits sounded so good.

His personal mathematical expertise and ability was perhaps
not fully developed at this time, as he confesses in several posts,
that he is not up to the math, and indeed enlists a friend
to assist him piece together several equations to describe
a number of his more exotic circuits.

In any case, John Broskie must be commended for taking the
brave and daring step of publishing his circuits and openly
sharing this thoughts, leaving him open to criticism.

If we move on toward John's own explanation,
we see several assumptions, reductions, simplifications present:

(1) The first stage is treated as a resistor-divider network,
as an explanation of how the Power Supply hum/noise enters,
and is found in the 1st stage output.

(2) The noise is assumed to come from the B+ power-supply.
For intents and purposes, John does not distinguish various sources of noise
in the first stage, but assumes it can be treated / removed simply.

(3) The hum/noise creeping in from both stages is assumed to be the same
as a copy of the noise that can tapped directly from the B+ supply.

(4) The amplitude of the noise signal is assumed to be semi-constant or linear
(or at least proportional), and so a complimentary copy can be bled off
from a real resistor-divider network across the B+ and fed to the following stage.

We will examine these assumptions / premises and axioms one by one.

 Okay, lets start with number 1:

(1) The first stage is treated as a resistor-divider network,
as an explanation of how the Power Supply hum/noise enters,
and is found in the 1st stage output.

At first sight, John's idea seems quite reasonable:
The tube and its Anode load are in series across the power supply,
and each is a resistance.
As a result, part of the B+ noise (hum, hiss) is dissipated
in the load, and part of this noise-signal appears across the tube.
If we reference ground, then it is the portion appearing across the tube
that we find on the output terminal and in the signal path.

The trouble is, at least some, perhaps a significant amount,
of PS noise is coming in from other directions:

(a) Heater-circuit to cathode bleeding. This source comes from another (often unrectified 60 cycle) output in a multi-part supply. Although it has a phase relation to the main transformer, there is no way of predicting its phase in relation to the 120 cycle full wave B+ supply. What is worse, two more factors come into play:

i) Its amplitude will go up and down with cathode current interactions, not in sync with the voltage dividing effect of the tube/load at the output.
ii) Its amplitude is likely to remain almost constant in comparison to B+ hum coming in through the Anode current.
iii) Its amplitude could be in any arbitrary proportion to the B+ sourced hum/noise.
iv) The "noise" component of the heater and B+ will have no correlation at all.

(b) Electromagnetic field hum pickup. This source again will not vary with input-tube anode current signal swings, except in as much as some of it may be caused by them directly. Most likely the bulk of such noise will be being injected into the 1st stage via the grid leads, independently of the amplitude of the input signal, and this constant source will be amplified by gain of the tube.

(c) Random Noise Produced in Various Components. This noise will be individual and unique to each circuit component, and cannot be accurately tracked or canceled by a similar type of 'white-noise' or 'pink-noise' generated in the B+ supply.

Even if John's method of injecting PS noise into the following stage was effective, it would only work for the hum component of the noise, and it would only work for a constant amplitude hum-component of the total hum in the first stage. Fluctuating hum components in the first stage cannot be mimicked by a resistor-divider network across the B+, unless the other fluctuating components are also being injected backwards into the Power Supply circuit in significant quantities!

This would be only the first caveat to the Voltage-Divider treatment of the first stage.
Noise from other sources in the 1st stage cannot be
tracked, copied or cancelled by the proposed method.