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In LA Vol 7 and 8 two very interesting articles were published by Mr. Giovanni Stochino
clarifying the subject of active feedback
The mathematics behind the various variants were elaborated in much detail, giving a good insight
how one system differed from the other.
Several implementations were given, illustrating how to translate a mathematical scheme into a
practical solution, and exactly here in the implementations it is that I have a different view on things.
I could not agree on two issues and several emails with the author could not bring us closer together.
The two points being:
1) To my opinion a BEF is just an AEF, and not something standing on its own
2) Using two resistors in the context of this article to form an adder is not at all a 1:1 replacement for
a mathematical adder.
1) AEF and BEF
The schemes given in said publication for an AEF and for a BEF, are shown below in figures 1 and 2.
Figure 1. Diagram of an AEF scheme.
Figure 2. Blok diagram of a BEF.
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Notice that figure 1 is a diagram with pure mathematical blocks, but that amplifier A2 in figure 2 is
already implemented with an op-amp. This magnifies the impression that a BEF scheme differs from
an AEF scheme.
.
However the direct path from Vs to Vo passing an adder and amplifier A1 is the same in both cases.
So to start with I will concentrate on the remaining parts S3 and A2 belonging to the AEF in Fig.1.
In three steps will l change-over carefully from the “mathematical” diagram in figure1 to a full blown
implementation of S3 and A2.
Step 1, A2 with the required bandwidth limitation fu2
In this first step I gave A2 the bandwidth limitation 1/2Pi*R1C1 = fu2 that it needs to have for
stability reasons when closing the loop around A1.
Step2, S3 implemented with an op-amp.
This second step shows how to implement S3 and A2 with real life op-amps.
Step3, two op-amps of step 2 merged into one.
And the third and last step shows the combination of Adder S3 and Amplifier A2 in just one op-amp.
Since Spice can emulate all steps, from mathematical modules onto practical implementations with
op-amps, I tested all versions to see if they are really identical in their transfer as well from the left
and the right side from the adder up to Vc, the output of A2.
In the article in Linear Audio, the main amplifier A1 had a (gain)-bandwidth of 10 Mhz, so the rest of
the circuitry has to exceed this bandwidth with a margin.
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For that reason I used an LT1806 for A2 and an LT1395 for the adder in step 2. I used 600 Ohm for all
resistors, including R1 and 25pF for C1, to restrict the bandwidth of A2 to the same 10 Mhz as A1,
just as required.
When driving the three variant from the left side, transfer is as shown in fig 3.
Figure 3: Transfer up to Vc for the three different steps driven from the left side of S3
Figure 4: Transfer for the three steps driven from the right side
It is obvious that all three variants are absolutely symmetrical in their behaviour when driven from
the right or from the left side.
However for transfer as well in amplitude as in phase, only step 1 and step 3 are fully identical. Step 2
gives almost the same amplitude transfer but the phase is falling more rapidly because of the extra
op-amp in the signal path.
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So step 3 is the preferred implementation solution, also because it has no connection to ground just
like the mathematical adder doesn’t have. With the used components, step 3 gives a 100% substitute
for a mathematical adder.
So coming from Fig.1, an AEF with this solution will move into figure 5 below.
Figure 5: AEF with a full implementation for S3 and A2 from Figure 1.
For the BEF from Figure 2, Ab is an RC network as explained In the article with the same
(gain)bandwidth as A1, and A2 has to be bandwidth limited as well to the same bandwidth.
That turns Figure 2 into
Figure 6. Taking the implementation of a BEF one step further from Fig 2.
Now where is the difference with the AEF in figure 5, well there is no difference.
As a confirmation that I did not miss anything, see figure 17 in LA Vol7.
The implementation of the bandwidth limitation around A2 is still missing in figure 17, but is
indicated between brackets next to A2 telling that Gain-bandwidth of A2 and A1 has to be the same.
A more detailed implementation is visible in in LA Vol7 Figure 19. The fact that you will see two
different time constants (600/25pF and 750/5pF) instead of two time constants of equal value, has
everything to do with the fact that the implementation of adder S from Fig. 6 above, is done with 2
resistors. This necessitates an amplifier A2 with a much larger bandwidth and changes the behaviour
of the circuit significantly as will be illustrated in my second point hereafter.
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Just for completeness, the transfer for a complete AEF including A1 but still with the mathematical
adder S4 (as in Fig 1. and in Fig.5) are shown below, again confirming that step 3 is a 100%
implementation for S3 and A2.
Figure 7, transfer of A1 (flat), and transfer for a complete AEF with resp. step1 and step 3.
Again, the transfer with step 1 and with the implementation in step 3 are exactly on top of each
other as well in amplitude as in phase, proving that both are 100% equal.
Not shown here, but in line with expectation, the tested step response and suppression of harmonics
are also perfectly the same between Fig. 1 and Fig. 5.
To conclude:
To my opinion what is called a BEF in the article is just a correctly implemented AEF.
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2)
The use of 2 resistors to realise adder S4 in Fig 1 with far reaching
consequences.
To explain my second point, I will again start with an AEF.
Figure 11 Diagram of an AEF (Aef1)
The output of adder S4 in Fig. 11 is equal to (Vs + Vc).
Substituting S4 by 2 resistors gives the diagram in Fig. 12 below.
Figure 12 AEF with a resistor adder for S4
In order to get the same (Vs + Vc) out of this resistor adder, the output of A2 should now be
different. The math behind it to get the same output is {(Vs) + (Vs + 2*Vc)}/2 = (Vs + Vc).
So to get the same Vo from A1, A2 should now produce (Vs + 2*Vc) instead of just generating Vc
To produce component Vs at least one decade beyond the bandwidth of A1 , can only be done when
no filter is in the “left” side of S3, so here is a discrepancy.
To build a perfect AEF, an R/C filter as pictured in in the left leg of Fig 5 is needed, but using a 2
resistor adder seems to point firmly in a different direction.
A Spice simulation will give the answer to this question.
I will therefore look at the transfer of the AEF implementation as pictured in the article and shown in
Fig 13 below. The implementation has indeed a missing R/C network in the left leg and as we know
now, it is missing for a reason.
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Figure 13. AEF implementation as in figure 12 in LA Vol7 (Aef2)
Negative aspects of using a resistor adder, are that A2 has now to produce 2*Vc, and
because of that the control range will at least be halved, but also the bandwidth of A2 has to
be twice as high.
So why not connect the output of A2 directly to A1, since A2 already has to produce both
signals Vs and Vc, thereby making S4 obsolete.
A2 will then only have to produce (Vs + Vc) instead of (Vs + 2Vc), making the control range
twice as large as in Fig 13.
And steering just A1 is quite a bit easier as it is to steer a low ohmic resistor adder, so
demands on A2 will be way more relaxed. Notice however that Figure 14 presents a pure FDR
(Feedback Distortion Reduction) and not an AEF scheme.
Figure 14 AEF without S4 (Aef3)
The proof of the pudding is in the eating, so I have simulated in Spice all the above situations called
resp. Aef1 to Aef3.
I used again an amplifier with a gain of 1 for A1 and with an fc1 of 10Mhz.
Bandwidth for A2 was set at 10Mhz for resp. Aef1 and Aef3, but at 20 Mhz for Aef2 because of the
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attenuation of 6dB by the resistors adder.
The resistor adder in Aef2 consisted of 2 resistors of 200 Ohm each.
Figure 15 Transfer function for all 3 versions Going from upper to lower curve at 100 Mhz, we
have resp. the transfer of A1 only, Aef1, Aef3 and Aef2.
All versions are comparable in amplitude response up to 30 Mhz, but only Aef1, the real AEF,
manages to stay transparent beyond this frequency and to eventually coincide with A1’s response
curve beyond 100Mhz, the transparency that sets an AEF apart from FDR as mentioned on page 74 as
the fourth requirement.
This is of course reflected in the phase behaviour, where only Aef1 does hardly add phase to the 90
degrees of A1 at 100 Mhz.
The other two versions Aef2 and Aef3 are showing almost equal but much larger phase distortion up
to 100 Mhz.
Several conclusions can be made from Fig 15:
1)
when using two resistors to implement Adder S4 in Fig. 11 the time constant in the “left leg”
should be discarded because Vs should be available at all frequencies.
2)
The FDR does even perform slightly better as an AEF scheme with resistor adder. This being
true both in Amplitude as in Phase. On top of that it has twice the control range and has a
relaxed job in driving A1.
3)
Because Aef2 and Aef3 perform identically, this leads automatically to the conclusion that an
AEF with a resistor adder for S4 is not performing as an AEF but like an FDR. This solution
does not seem to bring any benefit at all over a FDR.
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