Where Does Feedback Actually Reduce Distortion?

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here. 

Negative feedback is often described with a simple rule:

Distortion generated inside the feedback loop is reduced by the loop gain.

This statement is true, but it is incomplete. It leads to a common confusion: if a nonlinear element is hidden inside the overall forward transfer function, shouldn't feedback reduce its effect? And if a nonlinear element at the summing junction is "inside the loop," why is it said that feedback cannot correct it?

The resolution comes from looking at disturbances as equivalent input errors.

A general feedback system

Consider a nonlinear amplifier with feedback:

 

• $x$ is the input signal,
• $y$ is the output,
• $\beta$ is the feedback factor,
• $S$ is the summing element with transfer function $S(x,-\beta y)$,
• $e=S(x,-\beta y)$ is the signal at the output of the summing element,
• $G$ is the forward amplifier with transfer function $G(e)$, so $y=G(e)$.

Note that for the usual ideal summer, the transfer function is simply $S(x,-\beta y)=x-\beta y$, but a real summing element may be nonlinear and/or add terms unrelated to either $x$ or $-\beta y$. 

Now suppose the summer characteristic changes slightly:$$S\rightarrow S+\delta S.$$

Perhaps our summer is a differential amplifier (such as a long-tailed pair a.k.a. LTP), and its operating point changes slightly, or maybe we now have noise coming into the summer. How does the output change?

The perturbation of the summer creates an error $$\delta e=\delta S-\beta S_y\delta y,$$ where $$ S_y=\frac{\partial S}{\partial(-\beta y)}.$$ Let $g=G'(e)$ be the incremental (i.e. small signal) gain of the amplifier $G$. Then the output change is $$\delta y=g\delta e=g(\delta S-\beta S_y\delta y).$$
Solving for $\delta y$ gives us $$\boxed{\delta y=\frac{g\delta S}{1+\beta gS_y}}.$$The denominator contains the familiar loop-gain term, so at first glance, this appears to say that even summing-junction distortion is corrected by feedback. But this interpretation is incomplete.

The equivalent input error

Suppose instead that the input itself changes by a small amount $\delta x$. The corresponding output change is $$\delta y=\frac{gS_x}{1+\beta gS_y}\delta x,$$where $$S_x=\frac{\partial S}{\partial x}.$$Now define an equivalent input disturbance $\delta x_{\rm eq}$ that produces the same change in the summer output:$$S_x\delta x_{\rm eq}=\delta S.$$Therefore,$$\boxed{\delta x_{\rm eq}=\frac{\delta S}{S_x}},$$and the output disturbance becomes$$\delta y=\frac{gS_x}{1+\beta gS_y}\delta x_{\rm eq}.$$But this is exactly the same transfer function as for the desired input!

In other words, the feedback loop cannot distinguish between:

• a real change in the input signal, and
• a nonlinear error created in the summing element.

Both appear at the same location and are processed identically.

Why downstream distortion is different

Now consider distortion generated after the summing junction, for example in the output stage of an amplifier:$$y=G(e)+d.$$The disturbance $d$ is not present at the input of the forward amplifier. Linearizing,$$
\delta y=
\frac{d}{1+L},
$$where $L=\beta g {S_y}$ is the loop gain.

This looks like feedback has suppressed the distortion. However, to compare it with the input signal, we must refer it back to the input. The equivalent input disturbance is$$\delta x_{\rm eq}=\frac{d}{g {S_x}}.$$The large forward gain between the summing junction and the distortion source has already reduced the disturbance when expressed as an input error.

Feedback is not suppressing the equivalent input disturbance. It is simply reproducing it through the same closed-loop transfer function as the desired signal.

The general rule

Any disturbance can be converted into an equivalent input disturbance. The size of that equivalent input disturbance depends on where the original disturbance occurs.

For a disturbance generated after a forward gain $P$,
$$
\delta x_{\rm eq}
\approx
\frac{d}{P}.$$For a disturbance generated at the summing junction, $P=1$, so $$\delta x_{\rm eq}=d.$$This is why the summing junction is special. Not because feedback somehow stops working there, but because there is no forward gain before the error is created.

A better way to state the feedback principle

Instead of saying:

Feedback reduces distortion generated inside the loop.

a more precise statement is:

Feedback reduces the effect of disturbances according to their equivalent value when referred to the summer's input. 

It follows that

Any disturbance already present at the summer's input is indistinguishable from the input signal and receives no correction.

This viewpoint resolves several apparent contradictions in feedback theory:

• amplifier gain errors are reduced by feedback;
• distortion generated in later stages is reduced according to the gain preceding the distortion source;
• summing-junction nonlinearity is not reduced relative to the input signal;
• feedback never knows whether a signal at its input is "wanted" or "unwanted."

The loop only acts on the error signal it sees. Everything else is bookkeeping.

Zeno, Feedback Loops, and the Difference Between a Phenomenon and Its Description

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

Zeno's paradox of Achilles and the tortoise is usually presented as a puzzle about infinity. Achilles gives the tortoise a head start. To overtake it, he must first reach the tortoise's starting point. By then the tortoise has moved ahead. Achilles must then reach that new point, by which time the tortoise has moved again, and so on.

The modern mathematical resolution is familiar. If the tortoise is slower than Achilles, the time required to reach each successive point forms a geometric series:

$$
t = t_0 + rt_0 + r^2 t_0 + r^3 t_0 + \cdots
$$

where $0 < r < 1$. The sum is finite:

$$
t = \frac{t_0}{1-r}.$$

Calculus resolves the apparent contradiction.

What interests me is not the mathematical resolution itself but the source of the intuition that creates the paradox in the first place.

The physical event is straightforward: Achilles runs continuously and eventually overtakes the tortoise. The paradox appears only after an observer chooses to describe the event as an infinite sequence of checkpoints. Once that description has been introduced, it becomes tempting to reason about the checkpoints as though they were the mechanism of the motion itself.

The distinction is subtle. The checkpoints are real in the sense that they correspond to real positions. However, the decomposition of the motion into an infinite sequence of tasks is introduced by the observer. Achilles is not aware of the decomposition. The decomposition is a property of the analysis rather than a property of the runner.

A similar phenomenon appears in discussions of negative feedback amplifiers.

Consider the familiar error transfer function:

$$\frac{1}{1+A\beta}.$$

When $|A\beta|<1$, it can be expanded as

$$ 1-A\beta+(A\beta)^2-(A\beta)^3+\cdots.$$

The terms naturally suggest a story. An error appears at the output. A correction travels around the feedback loop. The correction is itself corrected. The process repeats indefinitely. The mathematics seems to support the intuition because the series resembles successive trips around the loop.

The difficulty is that useful negative feedback generally requires $|A\beta| \gg 1$, precisely where this expansion does not converge. The "error circulating around the loop" picture is therefore not a valid description of the physical operation of a practical amplifier.

For large loop gain, a more useful expansion of the same error transfer function is

$$ \frac{1}{1+A\beta} = \frac{1}{A\beta} \times \frac{1}{1+{1 \over {A\beta}}} = {1 \over {A\beta}} \times [1-{1 \over {A\beta}}+{1 \over {(A\beta)^2}}-{1 \over {(A\beta)^3}}+\cdots],$$

which converges when $|A\beta|>1$.

This expansion suggests a completely different intuition. Instead of a large error repeatedly corrected by the loop, it describes a small residual error with progressively smaller finite-gain corrections. Note that the physical amplifier has not changed. Only the mathematical representation has changed.

This observation suggests a possible common structure behind the two examples. In both cases, a phenomenon admits multiple mathematically valid descriptions. One description is then unconsciously promoted from a tool of analysis to an explanation of the mechanism itself. For Zeno, continuous motion is replaced by an infinite sequence of checkpoints. For feedback, a closed-loop equilibrium is replaced by an infinite sequence of corrections.

Neither decomposition is wrong. In fact, both are useful. The problem arises when reasoning shifts from the original phenomenon to the decomposition without noticing the substitution.

Perhaps this is why Zeno's paradox remains interesting even after its mathematical resolution has been known for centuries. The paradox may tell us less about motion than about a recurring habit of thought: once a decomposition becomes sufficiently natural, it becomes difficult to remember that it is a decomposition at all.

Importance of low distortion

When talking about the distortion of audio electronics such as a DAC or a power amplifier, low order harmonics, particularly 2nd and 3rd, are often considered "harmonious" and "benign". In contrast, high order harmonics, which are not harmonically related to the fundamental tone, are considered noxious. This argument usually supports the idea that tube amps sound better, because tubes only add low order distortion, or that low- or zero-feedback amplifiers sound better because feedback adds higher order harmonics.

Let's conduct a little experiment. Let's say we test an amplifier that adds 0.01% of the 2nd harmonic (H2) and 0.001% of the 3rd (H3):

The distortion is not particularly high and is all low order, harmonious and benign, perhaps even euphonic.

Now let's play a simple chord, A-C#-E:

Oops. In addition to H2 and H3, our benign test amplifier sputters a bunch of intermodulation products, musically unrelated to the chord, with levels comparable or in some cases above those of H2 and H3. These are not euphonic and would not be masked by music. Worse, they will mask the music itself.

So, an amplifier with low order and relatively low level, "benign" distortion may be euphonic with a single tone but not so euphonic with a chord. Practically speaking, such an amplifier would have genre preferences: it might sound fantastic on simple music such as solo vocal, but would get confused with anything moderately complex, and it would mush a full orchestra or, say, a Rammstein recording.

Current Dumping: Fine Print

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

In the previous post, I explained how current dumping works on an intuitive, qualitative level. Let's go into the next level of detail.

A current dumping amplifier is a feedback amplifier consisting of a low distortion, low power amplifier A and a high-power, distorting buffer ("current dumper", here Tr1 and Tr2)  inside a common negative feedback loop:

Negative feedback reduces, but not completely eliminates, the distortion that the current dumper adds to the signal. From my earlier post, the share of that distortion that remains at the output (point D at the schematic above) after the negative feedback is applied is given by the Error Transfer Function $${ETF={1 \over {1- A \times B}}}$$ where $A$ is the transfer function (basically, the frequency-dependent gain) of the integrator composed of A,R1,C1, and $B$ is the transfer function of the feedback network. In this case, the feedback network has unity gain and is connected to the inverting input of the integrator, so $B=-1$, and $${ETF={1 \over {1+ A}}}$$ Therefore of the total open loop distortion $\epsilon$ of the current dumper, at point D we observe $$\epsilon _D={\epsilon \times ETF}=\epsilon {1 \over {1+ A}}$$ Since the current dumper adds distortion $\epsilon$, for this to happen, the input of the current dumper (point A at the schematic above) should see "pre-distortion" $$ \epsilon _A = {\epsilon_D - \epsilon}={{\epsilon {1\over {1+ A}}}-\epsilon}=-\epsilon{A \over {1+ A}}$$ At the load, $\epsilon_A$ and $\epsilon_D$ combine in reverse proportion to the impedances of R2 and L1: $$\epsilon_{LOAD}\propto {\epsilon_A Z_{L_1} + \epsilon_D Z_{R_2}}$$ Note that the load impedance affects the absolute level of the combined signal at the load but not the proportion of $\epsilon_A$ to $\epsilon_D$.

Combining the last three equations and dropping the common denominator $1+A$: $$\epsilon_{LOAD}\propto {\epsilon (Z_{R_2} - A  Z_{L_1})}$$

Clearly, the perfect cancellation of the current dumper's distortion occurs when $$Z_{R_2} = A  Z_{L_1}$$

For the implementation above, under ideal conditions, $Z_{R_2}= {R_2}$, $Z_{L_1}= s {L_1}$, $A=1/{(s R_1 C_1)}$, and the perfect cancellation means

$$R_2={L_1 \over {R_1 C_1}}$$

which is the result from [1]: "For the linearity of Tr1 and Tr2 to be immaterial then L must equal RRC".

However, by going through the algebra above, we obtained a more general and quite remarkable result: a perfect distortion cancellation requires the ratio of $Z_{R_2} /  Z_{L_1}$ to mimic the amplifier's loop gain $A$. This gives us the freedom to make current dumping work under less than ideal conditions, as well as in different implementations than Walker's.

For example, the gain of an ideal integrator $A=1/{(s R_1 C_1)}$ is infinite at DC, which is practically impossible. A more realistic integrator has a transfer function $A=A_0/{(1+s{T_p})}$, where $A_0$ is the finite DC gain and $T_p$ is the time constant corresponding to the integrator's single pole. Substituting this into our perfect cancellation condition $Z_{R_2} = A  Z_{L_1}$, we obtain $${Z_{L_1}} = {R_2 \over A} = {R_2 \over A_0}{ (1+s{T_p})} = {R_2 \over A_0}+s {R_2 T_p \over A_0}$$

Since the impedance of a realistic inductor is $$Z_L=R_{ser}+sL$$where $Rser$ is the DC resistance and $L$ is the inductance, it is clear that ${R_2 T_p / A_0}$ corresponds to the inductance, and ${R_2/A_0}$, to the DC resistance of the inductor providing a perfect distortion cancellation.

Since we have five variables $L_1$, $R_{ser}$, $R_2$, $A_0$, $T_p$ and two equations $L_1={R_2 T_p / A_0}$ and $R_{ser}={R_2/A_0}$, any three variables can be chosen freely. Note that $A_0$ and $T_p$ together determine the unity-loop-gain frequency $F_0=A_0/(2 \pi T_p)$, which may be a more meaningful parameter.

Finally, let's make a quick reality check and see if the ideal distortion cancellation in this example can still be implemented with reasonable parts. In my previous post, I found that a perfect cancellation is possible with a L=3.3μH inductor, R2=21ohm resistor and an amplifier with the unity-loop-gain frequency F0 of 1 MHz. With the DC gain of the amplifier at, say, 80dB (×10k), the DC resistance of the inductor should be 21/10k=2.1mOhm for a perfect low-frequency cancellation. This looks too low to be implemented in real hardware, esp. after the impedance of PCB traces, etc., is taken into account. One way to address this is to reduce the DC gain (and hence the loop gain at the frequencies below the amplifier's pole) while keeping the same unity-loop-gain frequency. Another would be to increase the unity-loop-gain frequency, which would require a proportional increase in R2 and Rser. However, practically speaking, the perfect cancellation at low frequencies may not be as important, as the amplifier has more loop gain there, and the feedback, rather than feedforward, will take care of the distortion of the output stage.

Stay tuned for an implementation of current dumping different from Walker's.

References

1. P. J. Walker and M. P. Albinson, "Current Dumping Audio Amplifier," presented at the 50th AES convention, March 1975.
2. S. Takahashi and S. Tanaka, “Design and Construction of a Feedforward Error Correction Amplifier,” JAES vol. 29, pp. 31-37, Jan/Feb 1981.

Current Dumping Revisited

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

Current dumping is a way of constructing a power amplifier where a low-power, low-distortion amplifier is used to correct the distortion of a higher-power, but less linear, amplifier ("current dumper"). The underlying assumption is that it is easier to construct a low power, low distortion amplifier than a high power, low distortion amplifier.

Current dumping was introduced by quintessential English audio company Quad and was used in a series of Quad's power amplifiers starting with the Quad 405. Quad's founder, P. J. Walker, presented the concept at the 50th AES convention in 1975 [1].

There has been much interest and public discussion of current dumping in late 1970s and early 1980s. While most reviewers used more or less complicated math to explain why and how current dumping works, the basic implementation is easy to understand on an intuitive level.

The following schematic is from Walker's original AES paper:

Here, A is the low power, low distortion amplifier, and Tr1 and Tr2 form the "current dumper". The feedback for A is taken from the output of the current dumper. The load is connected to both A (via R2) and the current dumper (via L1). 

Any distortion appearing at the output of the current dumper Tr1 Tr2 (point D in the schematic) is fed to the load via two parallel branches:

• Via L1
• Via the integrator A R1 C1, followed by R2:

 

If these two branches feed the load with the distortion of equal amplitude and opposite in phase, the load will see zero distortion - this is the big promise of current dumping.

Intuitively, since L1's impedance increases with frequency at 20dB/decade, the distortion it feeds to the load falls with frequency at the same rate and lags in phase by 90°. The same distortion coming via the integrator also falls with frequency at 20dB/decade, has a 90° phase lag, and is inverted - that is, its the phase is opposite to the distortion coming via L1. Since the levels are proportional and phases are opposite, with the right choice of R2, the residual distortion from the current dumper can be nulled.

The circuit is particularly easy to analyze if the gain of A is assumed to be infinite. In this case, the inverting input of A (labeled F) is at the ground level (that is, zero distortion signal) due to the feedback via C1. If the distortion is nulled perfectly, the load (labeled L) is also at the ground level. The current via R1 is equal to that via C1, and the current via L1 is equal to that via R2. A little algebra quickly shows that this can only happen when the DC resistance of L1 is zero, and its inductance is L1=R1×R2×C1. In the language of the AES paper, "For the linearity of Tr1 and Tr2 to be immaterial then L must equal RRC".

It is worth nothing that the gain of the integrator A R1 C1 is the loop gain of the amplifier, and that R1×C1 is the time constant corresponding to the frequency where the loop gain becomes unity (in magnitude; a 90° phase lag remains). For a perfect distortion cancellation, the impedances of L1 and R2 at that frequency must be equal.

Let's make a reality check and see if the ideal distortion cancellation can be implemented with reasonable parts. A typical Miller-compensated audio amplifier behaves like an integrator A R1 C1 above and reaches the unity loop gain at the frequency of about 1 MHz (see my previous post for an explanation). 1MHz corresponds to R1×C1 of about 0.16 μS. (Walker's own values, shown on the schematic above, give the unity-loop-gain frequency of 4.8MHz, which I believe is rather optimistic with the parts that were available in 1975.)

A typical air core inductor of the type commonly found at the output of such an amplifier will have an inductance in the low single μH range, so let's use Walker's 3.3μH. With R1×C1=0.16 μS, the cancellation condition above gives us 21 ohm for the value of R2. If we want the whole current dumping amplifier deliver, say, 100W peak power (50W RMS on a sinewave) into a 8 ohm load, our low-power, low-distortion amplifier A would only need to provide about 140mW into R2 at 20kHz, and less at lower frequencies. That looks rather realistic.

What is unrealistic are the assumptions of infinite gain for A and a zero DC resistance for the inductor. Still, the cancellation condition can be generalized for a more realistic setup, but the details will have to wait for another post.

References:

1. P. J. Walker and M. P. Albinson, "Current Dumping Audio Amplifier," presented at the 50th AES convention, March 1975.
2. P. J. Walker “Current Dumping Audio Amplifier,” Wireless World, vol. 81, pp. 560-562, Dec. 1975.
   

Bootstrapped collector loads

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

A bootstrapped collector load is a pair of resistors connected in series, with their common point actively driven, often by a unity gain buffer via a large capacitor:

Bootstrapped loads have been popular in audio amplifiers for more than 50 years, although these days they are largely replaced by constant current sources.

The work of a bootstrapped load is easily understood. Because of the unity gain buffer, the right side of the capacitor $C$ (see the schematic above) sees the same potential as the bottom end of $R_2$. As long as the voltage across the capacitor is relatively stable, the voltage across $R_2$ is relatively stable, and so is the transistor's collector current flowing through $R_2$. In particular, the collector current changes only a little with changes in the collector potential (e.g. an audio signal), as if the collector load resistance is large.

Quantitatively, it is obvious that at DC, $R_1+R_2$ is the only load that sets the quiescent collector voltage. To find out AC response, one needs to e.g. write down node equations, which reveal that a bootstrapped load presents a frequency dependent impedance of the form $$Z = (R_1 + R_2) \times (1+s T_z)$$

where $T_z = (R_1 || R_2) C$.

The impedance increases with frequency at 20dB/decade, as would the impedance of an inductor. Effectively, a bootstrapped load is a synthesized inductor with the series resistance $R_1+R_2$ and the inductance $L = R_1 R_2 C$. For example, with R1=R2=1kOhm and C=10uF, the equivalent inductance L=10H.

In a real circuit, this synthesized inductor is not the only collector load - connected in parallel to it are the output impedance of the transistor and the input impedance of the buffer. If this input and output impedances are lumped into Ri:

then the combined collector load has the form $$Z = {R_i || (R_1 + R_2)} \times {{(1+s T_z)} \over {(1+s T_p)}}$$ where $T_z = (R_1 || R_2) C$ as above and $T_p = {{ (R_1 + R_2) \over (R_i +R_1 + R_2) } C}$.

I will look more into the behavior and practical uses of a bootstrapped load in separate posts.

High Precision Composite Op-Amps, Part 5 - Cart Before Horse

 

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

In my previous post on this topic, I gave a couple of practical examples of composite amplifiers with the topology described by John D. Yewen's article in Electronics & Wireless World, February 1987 and promised we can do even better.

One of the issues with Yewen's topology:

is that R1 R2 (in the schematic above) attenuate the signal amplified by U1, which causes U1 to work extra hard, or, more precisely, work with a higher input signal. As was discussed previously on this blog, this affects the linearity of U1's input stage and adds distortion that cannot be corrected by feedback.

One way to address this issue is to use a better opamp as U1, but it is much easier to move the divider to U1's input:

R1 R2 should be large compared to Ri to avoid an unwanted noise gain increase and a loop gain reduction (see the discussion in my previous post), but otherwise this works exactly the same as Yewen's composite, only with lower distortion.

Naturally, this approach also works with frequency dependent dividers:

High Precision Composite Op-Amps, Part 4 - A Practical Composite Chipamp with LM1875

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

In my previous post on this topic, I discussed the role of the voltage divider in John D. Yewen's composite op-amps (see his article in Electronics & Wireless World, February 1987) and improving the composite's loop gain at audio frequencies by making the divider frequency dependent.

Towards the end of the previous post, I promised that this approach works in hardware, too, so here are two practical examples.

The first is a plain Yewen composite with two dissimilar opamps - an OPA134 and an LM1875:

The divider (R4R5) attenuates the output of the OPA134 by 33/(2200+33)≅-36dB, which together with the opamp's GBW of 8MHz places the zero (see my previous post for an explanation) at about 130kHz. It is kind of low, but in testing, placing the zero at a higher frequency made for poor clipping performance. With such a divider, the OPA134 adds about 16dB of loop gain at 20kHz (more at lower frequencies), and this composite produces 0.003% THD at 1kHz, 20W into 8ohm, or about 1/6 of the distortion of a standalone LM1875.

The second example is a composite with the same opamps and a frequency dependent voltage divider C4C5R10:

The improved divider adds a pole at 2kHz and a zero at 150kHz, increasing the loop gain by another 16dB at 20kHz while maintaining stability and clipping similar to that of the first composite. With a careful PCB layout, this composite should be able to deliver 0.001% of THD at 1kHz, 20W into 8ohm.

Note that for both versions, the Zobel network (R6C2 and R11C6, respectively) with the values shown is required for stability.

Can we make the composite still better? Yes we can! Stay tuned...

High Precision Composite Op-Amps, Part 3 - More Loop Gain

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

As discussed in my previous post on this topic, the resistive voltage divider in John D. Yewen's composite amplifier (see his article "High-precision composite op-amps" in Electronics & Wireless World, February 1987):

adds a zero to the loop gain, which helps to achieve stability at the expense of the loop gain:

For audio applications, it is desirable to maximize the loop gain, at least in the audio band, but preserve the phase margin. One way to keep that zero and maximize the loop gain at audio frequencies is to make the voltage divider frequency dependent, for example:

Adding an inductor in parallel to R3 adds a pole-zero pair (disregarding the inductor's own series resistance, the pole is at $F_p={1 \over {2 \pi}} {{R_3 || R_4} \over L_1}$, the zero at $F_z={1 \over {2 \pi}} {R_3 \over L_1}$). With the values shown, we get about 12dB of extra loop gain at 20kHz with the same phase margin as without the inductor:

 A 22mH inductor may not be very practical, but a similar effect can be achieved with a resistive-capacitive divider, for example:

Here, the pole is at $F_p={1 \over {2 \pi R_5 (C_1 + C_2)}}$, the zero at $F_z={1 \over {2 \pi R_5 C_1}}$. With the values shown, the loop gain is about the same as with the inductor above:

 

Not bad for one additional passive component. It works in hardware, too - I will show a practical example in my next post.

High Precision Composite Op-Amps, Part 2 - Divide and Conquer

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

My previous post on this topic was on composite opamps from by John D. Yewen's article "High-precision composite op-amps" (Electronics & Wireless World, February 1987):

Appropriately choosing the voltage divider R1R2 at the output of U1 allows to achieve stability (obtain sufficient gain and phase margins) of the composite at the expense of the loop gain. Here, orange traces are the loop gain of the composite from the schematic above, while blue are the maximum possible (whether stable or not) loop gain with the same two opamps:

How can a simple resistive divider make the composite stable? Let me look at the role of the voltage divider in Yewen's composite.

Referring to the schematic above, U2 sees a (differential) input signal that is a sum of (i) the signal at the non-inverting output, where Ri and Rf connect, and (ii) the same signal amplifier by U1 and divided by R1R2.

At low frequencies, U1's gain is large, and U2's input signal is effectively that at its non-inverting input. The loop gain is the product of that of U1 and U2 and falls with frequency at 40dB/decade. At high frequencies, U1's gain is small, and U2's input signal is effectively that at its inverting input. The loop gain is just that of U2, falling at 20dB/decade.

The transition from "low" to "high" frequencies is a zero in the composite's loop gain, located at the frequency where the signal magnitudes at the non-inverting and inverting inputs of U2 are equal - that is, when the gain of U2 followed by R1R2 is unity. For a single-pole U1, that frequency is a fraction of U1's Gain Bandwidth Product (GBW):$$F_{zero}=GBW_{U_1} \times {R_2 \over {R_1+R_2}}$$In the example above, GBW is 10MHz, the divider's attenuation is 22, so the zero is at ${{10 MHz}\over 22} = {455 kHz}$.

That is, Yewen's voltage divider sets the frequency of a zero in the composite's loop gain. The higher the attenuation in the divider, the lower is the zero, and vice versa.

By the way, it is possible and quite practical to replace the fixed divider with a trimpot and adjust the zero to one's liking, e.g. to obtain the necessary phase margin.

Can we make it still better? Yes we can! Stay tuned...

Omicron Headphone Amplifier: Circuit Design

As I mentioned in the previous post, the idea was to build a compact, inexpensive, easy to build low distortion headamp.

One can do compact with discrete SMT circuitry, but that would not be easy to build, hence Omicron is built with opamps.

Common, easily available, inexpensive opamps may or may not be able to drive 32ohm cans directly, so Omicron needs a current booster. An integrated high-speed buffer such as BUF634, LME49600 or LT1010 would do the job, but you can get 10 opamps for the price. So the booster is a classic push-pull emitter follower - just two transistors.

In a circuit like that, there are two major sources of distortion under designer's control. The obvious one is the output stage, where transistors experience large variations of voltage and current, leading to nonlinearity and to distortion. Since we only need 100mW - that's 80mA peak into 32ohm or 11V peak into 600ohm - and it is not a battery-powered headamp, Class A or AB with sufficient quiescent current is an easy choice. Unfortunately, even Class A by itself does not deliver the distortion we have in mind, so we will place it inside a feedback loop.

Quite a few headamps use a current booster inside the feedback loop of an opamp. It works well at low frequencies, below 1kHz or so, but at higher frequencies the opamp's gain goes down, and distortion goes up. This produces familiar looking charts:

and familiar sound: feedback redistributes distortion to upper audio frequencies, adding brightness, making sibilants unnatural and muddling the midband with intermodulation products. We want to avoid this trap and push distortion below the noise level using more loop gain. So instead of a single opamp, Omicron uses two (two half of one dual opamp) in each channel.

The other source of distortion is the opamp's input stage. Its has a delicate job of comparing a portion of the amplified signal with that from the source, and its own distortion is indistingushable from the useful signal. Better opamps may have more linear input stages, but are not necessarily common, easily available or inexpensive. Thankfully, there is another way. Increasing the loop gain (which we need anyway) decreases the differential input voltage that the input stage sees, making its job easier and distortion - smaller.

That was for the differential component of the input voltage, but there is also the common mode component. In a typical non-inverting configuration, the differential voltage may be zero, but inverting and non-inverting inputs would be flying up and down with the full amplitude of the input signal. This generates measurable and audible distortion, which we want to avoid. So Omicron is an inverting amplifier.

Inverting, of course, reverses the absolute phase. There are many people who say they can hear absolute phase, and prove this by flipping the phase switch on their amp or reversing speaker connection. However, this is not a blind test and can be (and probably is) biased. A more subtle test was offered by Stereophile on their first Test CD:

Track 8 on the CD features an "absolute phase" demonstration. The sound starts out with its overall polarity one way around, but finishes with its polarity inverted. According to many writers, especially Clark Johnsen in his book The Wood Effect, the sound of human voice and many instruments will be more natural with the polarity correct—ie, so that an acoustic compression that reaches the microphone will be reproduced as an acoustic compression that reaches the listener's ear—than it will the other way. We have no idea which way 'round on Gordon's recording is correct, but as we have inverted the polarity somewhere in the middle, you will be able to hear for yourself if there is an audible difference between the two states. And can you identify where the change in polarity occurs?

If you believe that absolute phase is important, get the CD and listen to the track and find where the change in polarity occurs. If you can do it, then you may want to rewire your headphones 😉

So here is the actual schematic of one channel of Omicron:

Each channel of Omicron is a composite amplifier built with two halves of an NE5532 and a two-transistor complementary emitter follower (EF) as a current booster. There are two feedback loops - one global and one for the second opamp and the EF. This configuration was selected after comparing a number of alternatives, including a single opamp with a current booster, a Yewen style composite, a Samuel-Groner-super-opamp-style composite, and a few others.

Going left to right:

• R1R3R5C1C2 is the global feedback loop.
• R1C1 is the input LPF. The amplifier is relatively wideband, with -3dB point of its frequency response at several 100's kHz (measurements will follow).
• C2 provides lead compensation, improving phase margin.
• D3-D6 and R10 adjust the loop gain in case of clipping or slewing, helping recovery.
• R7R8R9C4C5C6 is a local feedback loop with two-pole compensation.
• Q1Q2 and associated parts are the output current booster. R13 and R14 set the quiescent current, D7 and D8 provide thermal compensation.
• L1R43R45C45 is the output filter that both helps stability with capacitive loads and protects Omicron from EMI ingressm from the headphones' cable. 
  

Gain is x2.5, which works for both low- and high-impedance headphones. Input impedance is relatively low at 2kOhm - modern sources will handle it easily. The amplifier does not include a volume control in order not to be tied to a specific part; a 10kOhm pot with linear characteristic can be used - together with the input impedance and the effect on gain, it produces a control characteristic which is closer to true logarithmic than many audio pots.

Stay tunes for measurements and construction details of Omicron.

Omicron Headphone Amplifier: Background and Intro

Omicron is a compact, low distortion headphone amplifier that I have been developing jointly with @Rus2000 from the RCL-electro.ru electronics forum. 

I wanted a good sounding headphone amplifier that would meet a few formal requirements:

• Distortion meaningfully - say by an order of magnitude - lower than that of a reasonable digital source, across the audio band. That means not more than -130dB of distortion.
  
• Suitable for headphones from 32ohm (i.e. Grado) to 600ohm (Beyerdynamic T1)
• At least 100mW of output power to deliver 120dB SPL peaks with the typical 100dB/mW headphones
• Stable with reasonable capacitive loads, say up to 10nF
  
• Common, easily available, inexpensive parts that will not disappear from the market tomorrow (or in a year)
• Usable by itself, without a hodgepodge of boards - that is, cross-feed, fast DC protection and turn-on delay and  EMI protection are all integrated on board
• Compact - doesn't need to fit into an Altoids can, but shouldn't be a full 17-inch box either
• Simple and easy to build, a comfortable weekend project

There are quite a few headamp projects around, yet it is not easy to find something that ticks all the boxes above. Low distortion and garden-variety parts seem to be two most difficult and often conflicting requirements. After looking around for some time, I decided to roll my own.

This is what we got:

• Outstanding linearity: Distortion better than 100 parts-per-billion (-140dB, 0.000 01%)
• Compact: Just one IC and two transistors per channel
• Inexpensive, commonly available parts (NE5532, BD139, BD140), easy through-hole construction
• Functionally complete: One 80x110mm board carries two channels, DC protection and an optional cross-feed circuit
  

Simplified schematic:

Some distortion measurements:

In the next post, I will discuss Omicron's amplifier circuit.

Which HF distortion measurement: IMD 19+20k or THD 20k?

Here is a (longish) quote from Bruno Putzey's post at the Audio Science Review forum which explains his preference.

Let's start by explaining why I prefer to measure distortion strictly inside the audio band. There is an ongoing controversy about whether signals above 20kHz might or might not be audible, but what is not controversial is that signals below 20kHz are much, much more audible than signals above 20kHz. So if you are in a situation where you have to choose between optimizing performance below 20kHz or above 20kHz, you go for optimizing the bit that we are most likely to hear. Even high-res enthusiasts seem to have tacitly accepted this a long time ago. Remember DSD? A DSD AD/DA converter that, when measured over 20kHz, would easily clock a SINAD of 120dB would "degrade" to 50dB as soon as you upped the measurement bandwidth to 40kHz. But what you heard was of course 120dB, the rest was for the bats. Note the delicious irony. DSD was hawked on the grounds that you needed >20kHz bandwidth for high fidelity, while its skyrocketing supersonic noise floor was excused on the grounds that it was inaudible. As it is, DSD is perfectly listenable. I can find no more eloquent argument that the ear is not very sensitive above 20kHz than DSD...

Anyhow, this is why I like to test amplifiers with test signals that in themselves would be audible (i.e. fit below 20kHz) and also read the distortion and noise only in the band below 20kHz. Of course, I know perfectly well that if you then do a THD versus frequency sweep, any readings above 10kHz are meaningless because even the second harmonic will be outside the audio band. But as our sensitivity to those harmonics drops off rather quickly around 20kHz (as does the ability of most speakers to reproduce them), it's fair to conclude that they do not say much about sound. On the other hand, we can't just go ignoring any underlying non-linearity. We still need to test for misbehaviour at high frequencies. If you choose to limit measurement bandwidth to 20kHz you have to include something like the 19kHz + 20kHz test. I didn't invent that procedure btw, I got that from Bruce Hofer at AP who recommends it. In fact his version is even neater, he uses 19.5kHz and 18.5kHz, making sure that even order products sit at even multiples of 500Hz (from 1kHz upward) while odd products sit at odd multiples of 500Hz, from 17.5kHz down, potentially fitting 37 distinct IMD products inside the band. This refinement doesn't make much difference with the 1ET400 amp of course since there aren't that many IMD products poking up over the noise floor.

Given the choice between a sinewave test at 20kHz which only produces inaudible products and a two-tone test that produces all kinds of in-band distortion I go for the latter. By implication, we should be designing a control loop that maximises loop gain all the way up to 20kHz, but not beyond. Any control system obeys a law called the Bode Inequality. This is the closest we control theorists have to mass-energy conservation. In the case of a class D amplifier it implies that if you maximise loop gain over a largeish fraction of the switching frequency you'll have to take it down really fast afterwards. So that's why the wideband THD vs frequency plot goes up somewhat suddenly at the end. It's a compromise I'm knowingly making. Consider the alternative: I could instead pander to the bat-eared crowd and choose to minimise harmonic distortion components up to 40kHz, say. That would mean accepting a lot less loop gain below 20kHz and hence higher distortion in the audible frequency range. It's not a good tradeoff.

Anyhow, this should explain why the high-frequency IMD spectrum is so much cleaner than a wideband THD test would lead you to expect. But as I see it, the former is the one that is most likely to have a meaningful correlation to sound.

(While I'm at it I ought to point out that the idle noise is noise shaped. This is visible on the broadband noise plots where you can clearly see the rise after 20kHz. The extra outband noise is caused by the comparator and driver chips and ends up being noise shaped by the control loop. I only wanted to mention that because the wideband THD vs F plots are mostly swamped by this HF noise which bore some explaining.)

High Precision Composite Op-Amps, Part 1

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

In my previous post,  I discussed how distortion can be corrected by cascading multiple opamps:

Obviously, in such a structure the stability problem arises. One possible solution was devised by John D. Yewen and described in his article "High-precision composite op-amps", published by Electronics & Wireless World in February 1987.

Yewen observed that, if every opamp in the schematic above is a perfect integrator (i.e. its open-loop gain falls with frequency at 20dB/decade), the closed-loop transfer function of the above arrangement is simple enough to calculate and to test for stability using the algebraic Routh-Hurwitz stability criterion, without ever looking at Bode plots. Further, the closed-loop transfer function can be manipulated to stability by adding voltage dividers at the outputs of the error correcting opamps:

As for choosing the dividers for stability, Yewen mentions the binomial series (1,1; 1,2,1; 1,3,3,1; ...) and the Butterworth coefficients (1,1; 1,1.41,1; 1,2,2,1; ...) and gives one worked example of a three-opamp unity-gain invertor.

Turns out, the general rule for Yewen's dividers is quite simple for:

• two- or three-opamp composite amplifiers...
• with the Butterworth coefficients...
• built from identical, single pole compensated opamps.

Say, we need an inverting amplifier with gain $(-A)$. For example, in Yewen's worked example above, gain is $(-1)$, so $A=1$. To achieve that, the feedback network $R1R2$ should divide the output signal by ${A+1} = {{R1+R2} \over {R1}} = 2$. Here is the rule:

• To build a two-opamp composite, add the opamp C1 and divide its output voltage by $(A+1) \times 2$.
• To build a three-opamp composite, add another opamp C2 and divide its output voltage by $(A+1) \times 4$.

For example, for an inverting composite with $A=2$, the following values will work:

All three circuits above are stable and have phase margin of about 65° (loop gain on the top, closed-loop transfer function at the bottom):

The price for such stability is diminishing returns on extra opamps - each additional error correcting opamp contributes less to the overall loop gain because of the voltage dividers. Also, the closed-loop transfer function peaks around the crossover frequency, so a low-pass input filter is required.

Does it work in practice? Yes, as long as the opamps are (i) identical and (ii) sufficiently close to single-pole, at least around the crossover frequency. Here is a practical composite with the negative gain of 10 and total loop gain of over 90dB, built from four OPA134:

It is quite stable and clips ok:

Can we make it even better? Of course we can. Stay tuned...

Hawksford's Error Corection and "Distortion Selector"

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

Hawksford's error correction (H.ec, see my previous post) compares the output signal of a (near-unity gain) output stage with that at its input and adds the difference back to the input, thus correcting the error $\epsilon$ that the output stage introduces:

H.ec is equivalent to the following structure, which sometimes is called a "distortion selector":

"Equivalent" here means that the two schematics above have the same signal transfer function $$STF={H \over {1-B+HB}}$$ and error transfer function $$ETF = {{1-B} \over {1-B+HB}}$$

Ideal compensation in H.ec (i.e. $B \rightarrow 1$) corresponds to an infinite loop gain ${B \over {1-B}} \rightarrow \infty$ of the distortion selector.

Naturally, both H.ec and the distortion selector can be applied not only to a unity gain output stage, but to any amplifier with gain $1 / K$; one just needs to scale appropriately the output signal:

If H is an amplifier with feedback, there is no need to use a separate scaler K; one can use the existing feedback network of H:

It looks a bit complicated, but one practical implementation is straightforward:

Here, A is the "main" amplifier, R1R2 is its feedback network, and C is the distortion selector. Two of the four voltage adders are inside A and C, respectively, and the remaining two are implemented with R1R2.

The distortion selector C by itself is not distortion free, which can be ameliorated by giving it its own distortion selector:

Along the same lines, one can add C3, C4 and so on.

Obviously, in such a structure the stability problem arises. One possible solution was devised by John D. Yewen and described in his article "High-precision composite op-amps", published by Electronics & Wireless World in February 1987. In my next post, I will have a closer look at Yewen's solution.

Hawksford's Error Correction (H.ec)

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

In 1983, Malcom Hawksford presented at the 74th AES convention a paper "Power Amplifier Output Stage Design Incorporating Error Feedback Correction With Current Dumping Enhancement". The idea was simple - compare the signal after the output stage with that at its input and add the difference back to the input:

On the schematic, H is the output stage with near-unity gain; $\epsilon$ is the error (distortion) introduced by the output stage; and B is the feedback network. Working from right to left: $$v_o = \epsilon + v_e H$$ $$v_e = v_i - v_c = v_i - B (v_o - v_e)$$ From the second equation we find $$v_e={{v_i - v_o B} \over {1-B}}$$ Substituting back into the first equation: $$v_o = \epsilon + v_e H = \epsilon +{{H(v_i - v_o B)} \over {1-B}} = \epsilon {{1-B} \over {1-B+HB}} + v_i {H \over {1-B+HB}}$$ That is, we have the error transfer function $$ETF = {{1-B} \over {1-B+HB}}$$ and the signal transfer function $$STF={H \over {1-B+HB}}$$

How the magic works: when $B=1$, $ETF=0$ and $STF=1/B=1$. Neither the transfer function of the output stage $H$ nor its distortion $\epsilon$ affect the output signal.

H.ec can also be analyzed as a positive feedback loop with the transfer function $1/(1-B)$ nested inside a negative feedback loop with forward gain $H$ and feedback gain $B$ - which gives the loop gain $H B / (1-B)$. It is equivalent to replacing $H$ with $H^\prime = {H/(1-B)}$.

One example of practical implementation is Thule Audio's Spirit IA-100 integrated amplifier. The simplified schematic is as follows:

Here, Q7 and R4 convert the difference between the output and input of the emitter follower Q2 Q5 into the collector current of Q7. R4 sets the transimpedance, that is, the rate of voltage-to-current conversion. Q7's collector current flows into the current mirror Q1 Q3 Q4 Q6. At the output of the current mirror, this current adds to the the current from the voltage amplification stage, represented by Ivas. The sum of the two currents is converted to voltage by R1 and becomes the input signal for the emitter follower. The value of R1 defines the proportion of the two currents at the summing point and hence $B$, so R4 needs to be adjusted for the lowest distortion. (The real IA-100 uses a push-pull emitter follower at the output and no copuling capacitor, but that is not essential for the discussion of H.ec.)

Another implementation, the paX amplifier by Jan Didden, published in Elektor Magazine in 2008, employs the AD844 instead of Q7 and the current mirror in the schematic above, but otherwise works the same.

Audio Amplifier Feedback - Estimating Poles in Lead-Lag Compensation Scheme

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

In the last post, I postulated that a feedback network combining lead and lag compensation:

has a transfer function with two poles and two zeros:$$B(s)={R_g \over {R_f+R_g}}{{(s T_{z1}+1)(s T_{z2}+1)}\over{(s T_{p1}+1)(s T_{p2}+1)}}$$where $T_{z1}=R_f C_f$, $T_{z2}=R_n C_n$, $T_{p1} \approx (R_f || R_g +R_n)C_n$ and $T_{p2} \approx (R_f || R_g || R_n)C_f$.

The actual transfer function, calculated from the circuit theory, is$$B(s)={R_g \over {R_f+R_g}}{{(s T_{z1}+1)(s T_{z2}+1)}\over{(s(R_f||R_g)C_f+1)(s R_n C_n +1)+s(R_f||R_g)C_n}}$$As usual, it can be transformed into many equivalent forms as needed. The exact poles are the roots of the quadratic equation $${(s(R_f||R_g)C_f+1)(s R_n C_n +1)+s(R_f||R_g)C_n} = 0$$It can be solved algebraically, but the result is unwieldy and obscures, rather than clarifies, the placement of the poles.

Instead, it is easier to estimate the poles as follows. The time constant of the pole associated with $C_n$ is calculated by assuming that $C_f$ is an open circuit at the frequencies of interest and that the signal source (here, the output of the opamp) has zero impedance, then computing the equivalent resistance “seen” by the $C_n$, which is $R_f || R_g +R_n$.

The time constant of the pole associated with $C_f$ is calculated by assuming that $C_n$ is short circuit at the frequencies of interest and, again, that the output of the opamp has zero impedance, then computing the equivalent resistance “seen” by the $C_f$, which is $R_f || R_g||R_n$. 

The approximation is based on a number of assumptions that I may look at in a future post, but is surprisingly accurate (within 1% of the actual frequency for a reasonable audio band network). 

In any case, the exact roots are not that important. The approximate formulas help to see what shapes the loop gain, and what values can be changed to optimize it. For an experienced designer, the Bode plot by itself becomes sufficiently informative to skip the formulas altogether.

Audio Amplifier Feedback - Combining Lead and Lag Compensation

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

The lead and lag compensation schemes can be combined in one feedback network:

The transfer function of this lead-lag feedback network has two poles and two zeros:$$B(s)={R_g \over {R_f+R_g}}{{(s T_{z1}+1)(s T_{z2}+1)}\over{(s T_{p1}+1)(s T_{p2}+1)}}$$where $T_{z1}=R_f C_f$, $T_{z2}=R_n C_n$, $T_{p1} \approx (R_f || R_g +R_n)C_n$ and $T_{p2} \approx (R_f || R_g || R_n)C_f$.

For given $R_f$ and $R_g$, both zeros and one pole can be placed freely, which makes the combined compensation scheme quite flexible. 

Compare the $1/B$ feedback factor for the lead-lag compensation scheme (blue) with both lead (red) and lag (light blue) alone; green is the open-loop gain: 

It is clear that unlike the pure lead compensation, the combined scheme doesn't extend the bandwidth; unlike the pure lag compensation, it preserves the loop gain in the audio band; and from the rate-of closure, it appears that the phase margin is similar for all three compensation schemes.

One possible design procedure for the combined compensation scheme is:

1. Choose $R_f$ and $R_g$ from the desired DC gain $(R_f+R_g)/R_g$ and the expected parasitic capacitances
2. Choose the time constants for the first pole ($T_{p1}$) and both zeros ($T_{z1}$ and $T_{z2}$)
3. Calculate $C_f=T_{z1}/R_f$
4. Calculate $C_n=(T_{p1}-T_{z2})/(R_f || R_g)$ 
5. Calculate $R_n=T_{z2}/C_n$

Audio Amplifier Feedback - Lag Compensation

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

Lag compensation involves an RC network connected between the inputs of an opamp:

It may be a bit difficult to understand, as feedback reduces the voltage across the compensation network, and thus reduces the effect of that network on the closed loop response. 

For the loop gain, however, lead-lag compensation does make a difference. Consider the following schematic: 

From the point of view of the feedback signal, the compensation network $R_N C_N$ is connected in parallel to $R_I$ and modifies the transfer function of the feedback network as follows:$$B={Z_I \over {Z_I + Z_F}}={{R_I \over {R_I + R_F}} \times {{s R_N C_N +1} \over {s (R_N + R_I||R_F)C_N +1}}}$$

The compensation network adds to the loop gain a pole at $\omega_p = {1 \over {(R_N + R_I||R_F) C_N}}$ and a zero at $\omega_z = {1 \over {R_N C_N}}$; note that $\omega_p < \omega_z$ (green is the forward gain of the amplifier, blue is $1/B$):

The loop gain is reduced by 3dB at $\omega_p$ and keeps falling until the zero cancels the effect of the pole, including the extra phase lag. The crossover shifts to a lower frequency, where the phase lag is smaller, increasing the phase margin (here, green is the loop gain without compensation, blue is the loop gain with lead-lag compensation):

The net effect of lag compensation is a reduction of the loop gain at higher frequencies without the phase lag that would normally be associated with such a reduction.

Compare lag compensation to dominant pole (e.g. Miller) compensation with the same bandwidth (green is the loop gain without compensation, blue - with lead-lag compensation, red - with dominant pole compensation):

Lag compensation allows more loop gain at lower frequencies while providing a very similar phase margin. Due to the finite gain of the amplifier, the closed loop frequency response with lead-lag compensation (blue) is slightly different from that with a dominant pole (red):

Sometimes in the lead-lag compensation network, either R of C is omitted. If R is omitted and C is left alone:

the feedback network transfer function becomes $$B={Z_I \over {Z_I + Z_F}}={{R_I \over {R_I + R_F}} \times {1 \over {s (R_I||R_F)C_C +1}}}$$ There is no zero anymore to compensate for the additional pole, and for stability, a zero would usually need to be introduced separately:

If C is omitted and R is left alone:

then the feedback network transfer function becomes $$B={Z_I \over {Z_I + Z_F}}={{R_G \over {R_G + R_F}} \times {R_1 \over {R_1 + R_G||R_F}}}$$ Since ${R_1 \over {R_1 + R_G||R_F}}<1$, the loop gain is decreased across all frequencies.

Useful links:

• Op Amps For Everyone by Ron Mancini
• Noise gain compensation (in Russian)
• Compensating Amplifiers That Are Stable at Gain ≥ 10 to Operate at Lower Gains
• APEX Application Note AN-20