## Friday, February 26, 2021

### Audio Amplifier Feedback - Lead Compensation

So far in this series of posts, I have been looking at shaping the loop gain of an amplifier by modifying its forward gain $A$:

Although I only scratched the surface of that topic, let me move to the gain of feedback network $B$.

Consider a hypothetical amplifier from my earlier post with three poles at 1kHz, 1MHz and 20MHz inside a feedback loop setting the ideal closed loop gain at 20dB:
As discussed in that post, the amplifier's accumulates just under 180 degrees of phase lag at the frequency where the loop gain is unity (crossover frequency), its phase margin is close to zero, and the amplifier is on the verge of becoming a generator:
Connect a small capacitor in parallel to the feedback resistor:
Although the forward gain (green curve) does not change, the ideal closed loop gain (the inverse of the feedback network's gain, blue curve) now has a kink:
Compare the loop gain with (brown curve) and without (red curve) the capacitor:
The capacitor extends the bandwidth from 4.4MHz to 6.7MHz and improves the phase margin from zero to 43 degrees. This is called lead compensation. Intuitively, the capacitor "speeds up" the feedback loop - extends its bandwidth and compensates some of the phase lag introduced in the forward path $A$.

Unlike dominant pole (e.g. Miller) compensation, lead compensation does not affect the loop gain at low frequencies, so there is no tradeoff between stability and loop gain, definitely a positive. On the other hand, where dominant pole compensation decreases the amplifier's bandwidths, lead compensation extends it, which may be problematic. As the zero-loop-gain frequency increases, high frequency poles add to the phase lag, the impact of parasitics increases, the variations in the parameters of parts with voltage and current become more important, the construction of the amplifier requires more attention, and so on.

The analysis of lead compensation can be found in many, many sources, such as Op Amps for Everyone by Ron Mancini, Operational Amplifiers: Theory and Practice by James K. Roberge, or this Burr-Brown application bulletin. The gist of it is that with the compensation capacitor, the loop gain becomes: $$LG=A \times B=A {R_2 \over {R_1+R_2}}{{s R_1 C + 1} \over {s(R_1 || R_2)C+1}}$$ The capacitor introduces a zero at $\omega_z={1 \over {R_1 C}}$ and a pole at $\omega_p={1 \over {(R_1 || R_2) C}}$; note that $(R_1 || R_2) < R_1$, so $\omega_z < \omega_p$. For the schematic above, the zero is at 2.85MHz, and the pole is at 28.5MHz.

Textbooks say that the properly placed zero should cancel one of the amplifier's poles. The reality is more complex. First, lead compensation only works when the crossover frequency (where the loop gain become unity) occurs between the pole and the zero:
That is, the value of the lead compensation capacitor cannot be chosen arbitrarily. While a larger Miller compensation capacitor always improves stability (at the cost of loop gain), the lead compensation capacitor must have just the right value. Too much or too little won't improve the phase margin, although it will still extend the bandwidth. In other words, a wrong value will brings all the costs of the extended bandwidth but no benefits of better stability.

Second, in a real circuit, selecting the value of the compensation capacitor is even more difficult because the forward gain and the crossover point are not fixed. With the signal, the parameters of various components change, the crossover point moves, and the lead compensation may become ineffective.

Third, a limitation specific to the lead compensation with a single capacitor is that, for a given closed-loop gain, the pole and the zero cannot be placed independently: $${\omega_p \over \omega_z}={{R_1 C}\over{(R_1 || R_2)C}}={R_1\over{R_1 || R_2}}={{R_1+R_2} \over R_2}={1+{R_1 \over R_2}}$$
That is, the zero and the pole are separated by a factor equal to the amplifier's ideal closed loop gain at DC as set by the feedback network. For an amplifier with 20dB gain, the pole will always be a decade higher than the zero. This fixed relationship is not always convenient.

The workaround for that last problem is to add a resistor in series with the capacitor:
The loop gain becomes $$LG=A {R_2 \over {R_1+R_2}}{{s (R_1 + R_3) C + 1} \over {s(R_1 || R_2 + R_3)C+1}}$$ and allows placing the zero and pole independently. In this example, the capacitance was decreased slightly to keep the zero at 2.85MHz, but the pole now is at 15MHz, not 28.5MHz as before.

Another version places the resistor differently:
but achieves a similar result: $$LG=A {R_2 \over {R_1+R_2+R_3}}{{s R_1 C + 1} \over {s(R_1 || (R_2 + R_3))C+1}}$$ Here a slightly higher capacitance was chosen to compensate for the smaller $R_1$ and keep the zero at 2.85MHz. The pole is at 14.3MHz.

## Saturday, February 20, 2021

### Audio Amplifier Feedback - LTP with LR compensation

Since my previous post was about shaping the open loop gain of a long-tail pair (LTP) with an RC network across the load, it seems to be a convenient moment to talk about LR compensation of an LTP:

The compensation network consists of an inductor $L$ in series with an optional resistor $R$, connected between emitters of the differential amplifier. Intuitively, at low frequencies where the impedance of the $RL$ network $Z_{RL}=R+sL$ is low compared to $R_e$, the $RL$ network reduces the degeneration added by $R_e$ and increases the transconductance of the LTP. At high frequencies, where $Z_{RL}$ is high compared to $R_e$, the $RL$ has little effect, and the LTP is degenerated by $R_e$:
To find out exactly how the transconductance changes in between these two extremes, I need to consider the small signal equivalent emitter circuit of the LTP:
Here, $r_e = {V_T \over I_E}$ is the emitter intrinsic resistance. The transconductance of the LTP with $RL$ compensation is $$g_m={1 \over {2 r_e+(R+sL)||(2 R_e)}}={{sL+(R+2R_e)} \over {2(R_e+r_e)(sL+R+2(R_e||r_e))}}$$ and has a pole at $s=-{{R+2(R_e||r_e)} \over L}$ and a zero at $s=-{{R+2R_e} \over L}$.

For example, the Bode plot above was generated with $R_e=220 ohm$, $R=0ohm$, $L=100 \mu H$ and $r_e={V_T \over I_E} = {{25mV} \over {1mA}}=25ohm$. With these values, the pole is at $s=-{{R+2(R_e||r_e)} \over L} = -{{0+2(220||25)} \over 10^{-4}} \approx -450k$, or $71.5kHz$, and the zero is at $s=-{{R+2R_e} \over L} = -{{0+2 \times 220} \over 10^{-4}} = -4.4M$ or $700kHz$.

One word of caution is that low $R$ in the $LR$ network undoes the degeneration added by $R_e$ and thus makes the LTP more susceptible to overload and nonlinearity, as discussed in the post of input stage linearity. The intended outcome is, of course, that the extra transconductance available with low $R$ increases loop gain, which reduces the differential input voltage seen by the LTP and improves its linearity. However, depending on the specific implementation, the loss of degeneration with low $R$ may become problematic.

Another potential problem is that the inductor is an effective antenna and will increase inductive coupling of noise from e.g. power supply rails into the LTP. The standard remedy is to use a shielded inductor and/or use two inductors in series connected in anti-phase, so that noise coupled into one is cancelled by the other.

## Tuesday, February 16, 2021

### Burning Amp BA-3b (Balanced) Updated Power Supply

In my previous post of the Burning Ampifier 3 Balanced, I mentioned that I replaced the CRC filters in the power supply with 100mF + 10mH + 100mF CLC filters per rail per channel. Here are some pictures of the new power supply:

From left to right, you can see two toroidal power transformers with a soft start mounted on top; four bridge rectifiers; first four capacitors, one per channel per rail, with their discharge resisotrs; four chokes; the other four capacitors; and the PCBs with the from ends, mounted on the rear wall of the chassis. On the far side, there are six pairs of power MOSFETs mounted on a heatsink. Here is a close-up shot showing how the capacitors and chokes are mounted:

The capacitors are Vishay/BCcomponents MAL2 101 16104, chokes are Hammond 159ZJ. The power transformers are from Antek.

## Saturday, February 13, 2021

### Audio Amplifier feedback - LTP with Frequency Dependent Load

In a number of previous posts, I looked at shaping the gain of a simple common emitter stage by applying frequency dependent local feedback. Local feedback, however, is not the only available gain-shaping tool. Another commonly used way is to add frequency dependent load.

Consider a simple long tail pair (LTP):

The analysis of the LTP can be found in many places online; my favorite is the ECE 3050 class notes by Marshall Leach. For small signals, the LTP is essentially a voltage dependent current source with transconductance $g_m={1 \over r_e} = {I_E \over V_T} = {I_{LTP} \over {2 V_T}}$ loaded with $R_1$ and $R_2$:

Clearly, by making $R_1+R_2$ frequency dependent, the gain $${v_o \over v_{diff}} = {{v_{C1}-v_{C2}} \over v_{diff}} = {{v_{diff} \times g_m(R_1+R_2)} \over v_{diff}} = g_m(R_1+R_2)$$ can also be made frequency dependent. For example, one can connect an RC network across $R_1+R_2$:

This modifies the load impedance from $R_1+R_2$ to $${(R_1+R_2)||({R_c+{1 \over {s C_c}}})}={(R_1+R_2){{(s R_c C_c+1)} \over {s(R_1+R_2+R_c)C_c+1}}}$$ with a zero at $\omega_z={1 \over {R_c C_c}}$ and a pole at a zero at $\omega_p={1 \over {(R_1+R_2+R_c) C_c}}$.

With the values shown, the zero is at 1.6MHz and the pole is at 2.65kHz (blue trace); for comparison, the green trace shows the response of the same LTP without the RC network:

In principle, the input impedance of the next stage (e.g. a common emitter amplifier) should be included into the calculation as connected in parallel to $R_1$ and/or $R_2$. That does not affect the zero, but the position of the pole can be more difficult to find. Note that the input impedance of a typical second stage of an audio amplifier with Miller compensation falls with frequency and can be quite low, pushing the pole to a higher frequency. This usually is a desirable outcome as it limits the phase lag introduced by the pole within the amplifier's bandwidth.

If the LTP is loaded not with resistors but with a current mirror, the formulas above still hold if you use the actual load impedance instead of $R_1+R_2$. The load would include the input impedance of the next stage in parallel with the output impedances of the differential pair itself and of the current mirror. For higher load impedance, the pole moves to a lower frequency. At the limit, if the differential pair is a pure current source with infinite output impedance, and the RC network is its only load, the pole appears at $w_z=0$.

## Saturday, February 6, 2021

### Audio Amplifier Feedback - NCore style compensation

After analyzing the two-pole compensation scheme with the C-R-C "T" network, it became clear that compensating an audio amplifier is not just about ensuring stability, but about shaping (usually maximizing) the loop gain in the audio range of frequencies. This can be achieved in many different ways, and here is another one.

At the 131st AES convention in 2011, Bruno Putzeys presented a paper titled High-Order Analog Control of a Clocked Class-D Audio Amplifier with Global Feedback Using Z-Domain Methods, discussing the details of then-new Class D amplifier technology, marketed by Hypex under the trade name NCore. An NCore amplifier has an analog control loop with five poles, one real plus two complex pairs. One pair comes from the LC output filter, and the others come from the following circuit:

An opamp is enclosed in a local feedback loop, part of which is the same C-R-C "T" network that is used in the two-pole compensation scheme. $R_f$ and $C_f$ add another pole-zero pair, and the transfer function of the whole circuit has the following form:
$$H(s)={v_o(s) \over v_i(s)}={A \over s}{(s+\omega_{z1})(s+\omega_{z2}) \over {s^2+{\omega_0 \over Q}s+\omega_0^2}}$$where $\omega_{z1}={1 \over {R_f C_f}}$, $\omega_{z2}={1 \over {R_t (C_1 + C_2)}}$, and $\omega_0=\sqrt{1 \over {R_f R_t C_1 C_2}}$.
That is, it includes an integrator (pole at $\omega=0$), zero at $\omega_{z1}$, a pair of poles with the corner frequency $\omega_0$, and another zero at $\omega_{z2}$, in this order. The positions of all poles and zeros can be chosen freely, although some values may be impractical given the parasitics.

For example, with the first zero at 72Hz, the pair of poles at 46kHz, and the second zero at 720kHz, the Bode plot looks like this:

Light blue is the Bode plot of an amplifier with NCore compensation network around its second stage. The gain set by the global feedback is the blue trace. As in previous posts, for comparison are also shown the same amplifier uncompensated (green) and with two-pole compensation (red).

This compensation network keeps the loop gain at about 55dB in the audio range of frequencies. (As I posted earlier, some experts believe that a flat loop gain across the audio range has a more palatable sonic signature compared to that of the loop gain falling at higher frequencies.) As the frequency increases beyond the corner frequency of the pole pair, the loop gain falls, first at 40dB/decade, then at 20dB/decade, ensuring sufficient phase margin and thus stability.

## Saturday, January 30, 2021

### Audio Amplifier Feedback - Two-Pole Compensation

As I noted in my posts on limitations of dominant pole compensation and on nested loops, dominant pole compensation locks a lot of gain inside the local feedback loop and makes it unavailable to the global feedback loop. The simplest way to unlock (part of) that gain is two-pole compensation.

Two-pole compensation implemented with a C-R-C “T” compensation network is well known since 1980s and was featured in Douglas Self's comprehensive article on frequency compensation in the February 1994 issue of Electronics World - see FIG 1D on the right:

An intuitive explanation of how the C-R-C compensation network works is simple:

• $R_P$ and $C_{P2}$ form a frequency-dependent voltage divider that feeds $C_{P1}$ with a portion of the output voltage
• At low frequencies, the impedance of $C_{P2}$ is high compared to $R_P$,  so $C_{P1}$ sees a only a small portion of the output voltage, the local feedback's loop gain is reduced, and more gain is available for the global feedback loop
• At high frequencies, the impedance of $C_{P1}$ and $C_{P2}$ is low compared to $R_P$,  so both capacitors in series work as a Miller compensation.
Between those two extremes, the amplifier with two-pole compensation has two poles (hence the name), so the loop gain falls at a rate of 40dB/decade, and a zero, which decreases both the rate of loop gain change by 20dB/decade and the phase lag. Compared to Miller compensation with the same bandwidth, two-pole compensation provides a welcome boost of global loop gain at audio frequencies:

Despite being around for 30+ years and probably because of its simplicity, a formal analysis of two pole compensation was published only in 2010 by Harry Dymond and Phil Mellor from University of Bristol, who presented their paper at the 129th AES convention. A brief summary:

• Similarly to Miller's single dominant pole, the location of the two poles depends on the amplifier's stage inside the local feedback loop
• The zero is located at the frequency $f_z = {1 \over {2 \pi R_P (C_{P1}+C_{P2})}}$
• Unity loop gain frequency is $f_0 = B \times g_m {(C_{P1}+C_{P2}) \over 2 \pi C_{P1} C_{P2}}$, where $B$ is the transfer function of the feedback network and $g_m \approx {1 \over {r_e + R_e}}$ is the transconductance of the input stage

Practically speaking, let $C_{P1}$ have the same value as for Miller compensation, choose $C_{P2}$ twice as big, and make $R_P$ as low as you can without affecting stability.

There is a so-called "output inclusive" version of the two-pole compensation where the bottom end of $R_P$ is connected not to the ground but to the output of the amplifier. The output-inclusive version has almost identical design equations to what is discussed above.

## Thursday, January 28, 2021

### Velleman K4040 Photo Gallery

This is one gorgeous looking amplifier!

## Saturday, January 23, 2021

### Audio Amplifier Feedback - Transient Intermodulation Distortion (TIM)

Now that I covered the linearity of the input stage, it is a good time to talk about TIM - Transient Intermodulation Distortion.

In 1970, Matti Otala published a paper called "Transient Distortion in Transistorized Audio Power Amplifiers". He argued that under certain conditions, fast signal transients create a specific type of distortion, which he called Transient Intermodulation Distortion (TIM).

In the paper, Otala considered a simple model amplifier with an RC compensation network providing a dominant pole:

Otala fed this model amplifier with a fast voltage step and observed something like this:
Here, the green is the step at the input $V_2$, blue is the slower output step, scaled down by the feedback network $\beta$, and red is $V_3$, the difference between the input and scaled output which is fed to the input stage $A_1$.

When $V_2$ grows faster (has a higher slew rate $SR=dV/dt$) then the compensation RC network can reproduce, $V_3$ overshoots its steady state level, as the red trace shows. Depending on the difference in slew rates at the input and at the output, the overshoot can be large.

Otala recognized that the overshoot at the input of the first stage may lead to the first stage clipping and thus to distortion - the effect he called Transient Intermodulation Distortion, or TIM.

I demonstrated in the previous post that a difference amplifier typically used as an input stage can easily be overloaded by relatively small signals, and that is becomes nonlinear and start distorting long before it clips. This is one of the mechanisms that potentially can give rise to TIM.

I have been arguing in my first post on audio amplifier feedback that the the difference signal seen by the input stage can be effectively reduced by increasing the loop gain. Indeed, the higher the loop gain is, the smaller both the overshoot and the steady state value of $V_3$ are, and the faster the overshoot settles to the steady state value:

The price of faster settling and smaller overshoot with higher loop gains, however, is the larger current though the RC network, all of which must be provided by the first stage $A_1$:

The extra current required by the compensation network may also lead to the first stage overload, but at the output rather than at the input. This is another mechanism that can potentially generate TIM.

The conclusions should have been that, in order to minimize TIM:

• The slew rate at an amplifier's input should be limited (by, for example, a low pass filter) to what the amplifier can reproduce at the output; and
• The first stage $A_1$ should be designed to both handle the difference input signals and drive the compensation network with low distortion; the now-standard approach (see my previous post) is to degenerate the input differential pair with emitter resistors and increase its tail current.

Otala came to the first conclusion, but not the second. Instead of designing a more robust and linear input stage, he set out to avoid both the overshoot at the input and the need to drive the compensation network at the output of $A_1$ by increasing the amplifier's slew rate, refusing to apply the dominant pole compensation and reducing the loop gain in the global feedback loop.

Importantly, Otala came up with the idea that there is an optimal level of feedback (loop gain), above which TIM would become the dominant distortion mechanism. The theme has become popular and led to the concept that it is the slew rate, and not the loop gain, that determines an amplifier's distortion. It still reverberates around the audio community in the form of amplifiers with "moderate feedback" and slew rates of 3kV per microsecond.

There is nothing wrong with high slew rate amplifiers. For a typical three stage Miller compensated amplifier, the slew rate can be an indication of how far the first stage of the amplifier is from clipping, and therefore how linear it is. It can also be an indication of the amplifier's bandwidth and hence of the loop gain in the audio range of frequencies.

Moderate feedback, on the other hand, is a problem. As I posted earlier, moderate feedback provides only moderate suppression of distortion and makes the input stage work harder and distort more. Conversely, high gain in the global feedback loop reduces the difference input signal for the input stage and thus reduces TIM. There is no need to choose between high loop gain and low TIM; one can - and should - have both.

## Saturday, January 16, 2021

### Audio Amplifier Feedback - Input Stage Linearity

In my previous posts, I mentioned that dominant pole compensation limits loop gain, which increases distortion in two ways.

1. Lower loop gain means that the feedback loop has lower ability to correct distortion as the Error Transfer Function $ETF={1 \over {1 - LG}}$ increases.
2. Lower loop gain means the input stage needs to deal with a larger signal.
In this post, I will focus on the second point and look at how lower feedback affects the input stage.

The input signal seen by the input stage of a feedback amplifier with loop gain $LG$ is $V_{diff}$, the difference between the input signal to the whole amplifier, $V_x$ and the portion of its output signal produced by the feedback network, $B \times V_y$ (see also the first post of the series): $$V_{diff}={V_x + {B \times V_y}} = Vx {1 \over {1 - LG}} + V_{err} {B \over {1 - LG}}$$
Clearly, the input signal seen by the input stage of a feedback amplifier depends on the loop gain - the larger the loop gain, the smaller the differential input signal.

For example, for an amplifier with 30dB (= approx 32) of feedback and 2V peak input voltage, the peak difference voltage the input stage sees would be (assuming the distortion is small compared to the signal and ignoring the sign) $$V_{diff}=Vx {1 \over {1 - LG}}= {2V \over {1 - 32}} = 65mV$$
Is that large or small? Let us look at a typical input stage - a differential amplifier a.k.a. long tail pair (LTP) - and at how its output voltage (between collectors) changes with the differential input signal (between bases):
It can only handle about 60mV peak input voltage before clipping, and is visibly nonlinear with much smaller inputs:
Oops. With 30dB of loop gain, the input stage will clip at relatively low input signals!

The standard way of fixing this is to degenerate the input stage by adding resistors into the emitters of the LTP:
The typical value of resistors is about 10x the intrinsic emitter resistance, which depends on emitter current $r_e={26mV / I_e}$. For the emitter current of 0.5mA, $r_e$ is about 50ohm, so 470ohm generation resistors are acceptable.

With degeneration, the gain decreases but the LTP can handle much larger input signals:

The nonlinearity is not so evident, so let's have a closer look:

This is 0.0074% of total harmonic distortion (THD). Unlike the distortion of the output stage, this distortion cannot be corrected by the feedback loop, as it is appears at the amplifier's input and is indistinguishable from the input signal. That is, 0.0074% will appear at the amplifier output.

Nevertheless, there are ways to decrease this distortion:
• One is to make the input stage intrinsically more linear. This usually means making it more complicated or, if the input stage is an opamp, selecting a better (and more expensive) opamp.
• Another is to make the differential input signal smaller by increasing the loop gain.
For example, with 100dB of loop gain, the distortion of a simple two-transistor differential input stage is unmeasurable:

With high loop gains, there is no need for complicated input stages or expensive opamps. Even a single transistor will do the job perfectly.

## Sunday, January 10, 2021

### Krell KSA-5 Clone - Revised Schematic and Build Guide

In the previous two posts on the topic, I discussed my approach to upgrading the KSA-5 and demonstrated the massive distortion improvement I was able to achieve.

Without further ado, here is the revised schematic:

The list of parts required to modify two channels:
1. 8x 0.22 ohm 2W or 3W resistors
2. 4x 22 ohm resistors
3. 6x 47 ohm resistors
4. 8x 100 ohm resistors
5. 8x 274 ohm resistors
6. 4x 332 ohm resistors
7. 2x 562 ohm resistors (not needed if you can reuse R5/R8)
8. 2x 3.92k resistors
9. 2x 1 Megohm resistors
10. 2x 68pF 50V capacitors, NP0/C0G ceramic or mica
11. 4x 4.7nF film capacitors
12. Some 18 AWG single core insulated wire and 2x 10 ohm 2-3W resistors for the output RL network

The list of changes to the original schematic:

1. Replace R1, R2, R6, R7 with 100 ohm resistors
2. Replace R5 and R8 with 332 ohm resistors
3. Replace R9, R10, R11, R12 with 274 ohm resistors
4. Replace R16 and R17 with 22 ohm resistors
5. Replace R19 with a 562 ohm resistor (reuse one of R5/R8)
6. Replace R23 with a 1 Megohm resistor
7. Replace R24 with a 68pF 50V NP0/C0G ceramic capacitor
8. Replace R33, R34, R35 and R36 with 0.22 ohm 2W or 3W resistors
9. Replace R37 and R38 with one 47 ohm resistor connected between bases of Q23/Q24 and Q25/26. Make sure to connect the new resistor correctly - see the build guide below - or your output transistors are at risk.
10. Replace R47 with a 3.92k resistor
11. Remove C2 and C3
12. Add a 4.7nF film capacitor and a 47 ohm resistor, connected in series, between the collectors of Q2 and Q3. Add another 4.7nF film capacitor and a 47 ohm resistor, also connected in series, between the collectors of Q7 and Q8. Place the new parts on the underside of the board if you like.
Repeat for the other channel. That's it!

As resistors's names are not marked on the board, here is a photo showing what to replace with what. Note that the additional parts from step 12 above are not shown - they are under the PCB. Click the picture to open it in full resolution.

After assembly, take the usual precautions before powering the amplifier up, as if it were a newly assembled board. Turn the bias adjustment trimpots all the way counterclockwise, connect a current limited +/-21V power supply with the current limit set at 0.5A per rail.

With power on, check the output for a possible oscillation, then adjust the bias and re-check for oscillations. The bias level needs to be at about 20mA per transistor - with 0.22 ohm emitter resistors, it corresponds to 4mV between the test points, which should be easy to measure with a DVM. Higher bias levels are possible, but the distortion will be slightly higher.

Let the amplifier warm up for 10-15 minutes and readjust the bias - it should go down as the output transistors warm up.

As with any feedback amplifier, capacitive loads may affect stability. In my testing, the amplifier remained stable with capacitive loads of up to 100nF. Consider adding the usual RL network between the output of each channel and the load to ensure stability. Make 20-30 turns of 18 AWG single core insulated wire on a 1/2 inch (12mm) former - a Sharpie will work - to make an air core inductor, then connect a 10ohm 2-3W resistor in parallel to it.

## Saturday, January 9, 2021

### Audio Amplifier Feedback - Nested Feedback Loops

In the previous post, I looked at how Miller compensation reduces the global loop gain of an amplifier. Where does the gain go? It gets trapped inside the local loop that the compensation capacitor forms around the second (transimpedance) stage.

When several stages are included within one feedback loop, the individual gains of each stage multiply (left). However, when one of the stages is enclosed in a local loop, the contribution of that stage to the global loop gain changes - it becomes the Signal Transfer Function set by that local loop (right).

With Miller compensation, the magnitude of the second stage's Signal Transfer Function falls with frequency by 20dB/decade. Because of that, at low frequencies most of the raw open loop gain of the second stage is applied to the global feedback loop. As the frequency grows, the gain of the second stage is increasingly removed from the global loop and locked inside the local loop.

As discussed in the last post, the decreasing global loop gain affects distortion at high frequencies. The effect is quite dramatic - here is an illustration from Douglas Self's Desiging Audio Power Amplifiers book:

In the next post, I will look into how loop gain affects the linearity of the input stage of an amplifier.