An experimental 4-th order linear audio power amplifier

[GK]

LOL, OK, thanks.... One listener is better than none, I guess. 


Here is the simulated versus measured response of the 5th-order loop to a square wave stimulus. Those familiar with 2-pole compensation would be aware of the compensation schemes inherent overshoot  to transient stimulus due to the necessary zero in the forward gain path. Well, a high-order control loop like this no different, just that the degree of overshoot is inherently greater due to the higher order of the control loop.

The first picture shows the (20kHz) square wave response to an non-bandwidth-limited square wave input signal. Just like with 2-pole compensation, the transient overshoot can be suppressed by bandwidth limiting the input signal. This is shown in the second picture. In this particular case, a low pass input filter comprised of a 1k resistor and a 1nF capacitor is sufficient to completely suppress the transient overshoot. Note that the time constant is in fact lower in the scope photo shown as the total resistance is 1600 ohms due to the 600-ohm output resistance of my signal generator (I also increased the simulation resistance to 1600 ohms to match the measurement).
   
A quite successful demonstration of basic control theory and simulation matching real-world implementation, methinks........







Oh..... another thing to mention - in the simulation schematic shown in my opening post it can be seen that the DC gain of the integrators was limited by the 1M resistors in parallel with the integration capacitors. I only needed to include these resistors to get the simulation to converge. They are not included in my real-life prototype. This is really an experiment in applying global loop feedback to a crazy degree without any form of pseudo-scientifically-justified hobbling ......... the prototype system is up and running 100% stable with hundreds of dB of open loop gain at DC (effectively the full open loop DC gain of the four series-connected LME49710 op-amps of the integrator stages plus another 100db or so of gain provided by the discrete power amplifier section).

[IanB]

I can happily report that, so long as the amplifier is operated within its linear region of operation (ie not driven into clipping), it is 100% stable in the 5th order mode of operation with any signal stimulus I have (exhaustively) tested it with.

A question here from a novice: does not amplifier stability depend on the nature of the load? What happens if you give it a strongly reactive and "difficult" load?

[megajocke]

Cool! 

[GK]

I can happily report that, so long as the amplifier is operated within its linear region of operation (ie not driven into clipping), it is 100% stable in the 5th order mode of operation with any signal stimulus I have (exhaustively) tested it with.

A question here from a novice: does not amplifier stability depend on the nature of the load? What happens if you give it a strongly reactive and "difficult" load?

A capacitive load can decrease the phase margin of the amplifier due to the pole formed by the load capacitance in conjunction with the finite open loop output impedance of the power output stage. If the amplifier has a marginal phase margin to begin with a moderate capacitive load can force it into instability. However that is not the case here. I've run it (square wave stimulus) into 2.2uF in series with 2.2 ohms and the amplifier happily continued as normal, burning out the resistor. That is without, at this stage, an off-board capacitive-load-isolating inductor directly at the output of the amplifier.


 

[GK]

Cool! 

Hey, thanks.


The control circuitry construction is coming along nicely, but I now need to give it a rest for the night. It looks like I'll be listening to tunes through this thing hooked up here in my study by late tomorrow evening........ stay tuned for some scope shots of stability-controlled clipping events that match the simulation results............. 

[robrenz]

I am still following GK,  I am hoping some of this will sink in by osmosis  

[c4757p]

I am still following GK,  I am hoping some of this will sink in by osmosis

Yeah, me too. This is quite interesting. I'm still trying to find a decent way to teach myself control theory like this - I can stabilize an SMPS feedback loop with some effort, but this is far beyond me. The SPICE/scope comparison shots are especially impressive. Clearly you know what you're doing.

[GK]

The SPICE/scope comparison shots are especially impressive. Clearly you know what you're doing.

Well, I think I know what I’m doing for the most part. I deleted my previous post as under further investigation I turned out to have made some incorrect assumptions. One thing that has me baffled right now, thinking about the whole project a bit more, is the systems complete stability to transient input signals that are too fast for the system as a whole to respond to, resulting in slew-rate limiting. This causes stages of the amplifier to saturate, temporarily. 

However, the temporary state of saturation during a period of slew rate limiting, both in SPICE and in the actual prototype, does not appear to be a trigger for instability of the high-order loop. So long as the 5th order system is operated within it linear limits (i.e. not clipping), no matter what kind of input signal I drive it with, I have been unable to excite any form of oscillatory burst at all – not just in SPICE, but, more importantly, in the actual real life realisation sitting on my bench top.

That “saturation” during clipping is (completely predictably) a trigger for instability while “saturation” during slew-rate-limiting doesn’t appear to have any negative impact on system stability at all is something that I do not fully understand at this point in time. This immunity of the system (stability wise) to “transient saturation” during slewing is demonstrated by the (non-bandwidth-limited) square wave response scope photos that I posted.

The experiment thus far leads me to believe that some of the theoretical “dangers” of conditionally stable control systems that I have read about are not necessarily true or at all insurmountable - the thing on the bench works!

[C]

Something to keep in mind.
Tweeter's will some times burn up from the increased energy generated in the high frequency spectrum when a signal clipped. The speaker systems I have seen do not expect a lot of energy to be in the tweeters frequency band.
If your amplifier chain  is capable or your limiter action has increased the high frequency energy enough, your amplifier may be fine, but the tweeter on the output may have died.

C
     

[GK]

Well sure, but that is true for any solid state power amplifier. One reason to use an amplifier that is powerful enough so that you are not regularly driving it into clipping.

[GK]

The SPICE/scope comparison shots are especially impressive. Clearly you know what you're doing.

Well, I think I know what I’m doing for the most part. I deleted my previous post as under further investigation I turned out to have made some incorrect assumptions. One thing that has me baffled right now, thinking about the whole project a bit more, is the systems complete stability to transient input signals that are too fast for the system as a whole to respond to, resulting in slew-rate limiting. This causes stages of the amplifier to saturate, temporarily. 

However, the temporary state of saturation during a period of slew rate limiting, both in SPICE and in the actual prototype, does not appear to be a trigger for instability of the high-order loop. So long as the 5th order system is operated within it linear limits (i.e. not clipping), no matter what kind of input signal I drive it with, I have been unable to excite any form of oscillatory burst at all – not just in SPICE, but, more importantly, in the actual real life realisation sitting on my bench top.

That “saturation” during clipping is (completely predictably) a trigger for instability while “saturation” during slew-rate-limiting doesn’t appear to have any negative impact on system stability at all is something that I do not fully understand at this point in time. This immunity of the system (stability wise) to “transient saturation” during slewing is demonstrated by the (non-bandwidth-limited) square wave response scope photos that I posted.

The experiment thus far leads me to believe that some of the theoretical “dangers” of conditionally stable control systems that I have read about are not necessarily true or at all insurmountable - the thing on the bench works!

Ahhhh haaaaa..........I think I'm starting to get a handle on this now............ For the high-order, conditionally-stable system to be triggered into sustained oscillation by a saturation event related to clipping, the saturation event itself must be of sufficient duration.

When the system is successfully triggered into global loop oscillation, the frequency of oscillation is confined to be somewhere within the range that the total phase shift around the loop exceeds 360 degrees. This frequency range is in fact not particularly high and lays quite a bit below the amplifiers unity loop gain frequency (the frequency at which the loop gain goes through unity at a single-pole roll-off).

It appears that the duration of the saturation event required to trigger global loop oscillation is roughly commensurate with the frequency at which oscillation can occur.

As it turns out, this duration is apparently orders of magnitude greater than the duration of time that any stage of the amplifier can actually remain in a state of saturation during slew-rate limiting.

If my preliminary observations as just detailed are correct, then they would explain why slew-rate limiting of the amplifier in response to fast rise-time input signals has so far been demonstrated to have absolutely no adverse effect on global loop stability; that is without the capacity to trigger any kind of observable oscillation, either temporary or sustained, of the high-order loop. This, if correct, is, I have to say, a very pleasing and reassuring observation indeed.

Is there a "light bulb" smiley avaliable anywhere? (that is with the light bulb above the head rather than sticking out of the mouth) 

[digsys]

<munching on popcorn and waiting for the bad guys to appear>

[robrenz]

There are signs of osmosis actualy working

[GK]

What bad guys?


I am now sitting here, listening to tunes through an audio power amplifier that:

1) Has a 5th order control loop
2) Is completely stable
3) Clips as clean as a whistle
4) Has approximately 650dB of open loop gain at DC
5) Has 64dB of loop gain at 20kHz (actually verified by measuring the amplitude of the virtual earth error signal)

I'm sorry to say it, but I really don't think that I would have a great deal of success trying to measure the amplitude of the error signal at 20Hz or so! 

I also have a big smile on my face right now.

Attached are photos showing the amplifiers clipping performance at 20kHz, 1kHz and 20Hz, along with a close up of the beast itself.
The yellow trace is the amplifiers output signal, driving a 4 ohm load. The blue trace is the 5V logic control signal to the analogue switches. When it is low, the system is switched into 1st order mode; 5th order mode when high.


The next step? To build a proper amplifier, with a decent PCB layout, capable of delivering a decent amount of power (100-200W).

[GK]

.........I'm sorry to say it, but I really don't think that I would have a great deal of success trying to measure the amplitude of the error signal at 20Hz or so! 

Just to put some kind of perspective on things here, this amplifier has a loop gain characteristic that rises at a 100dB/decade rate (5-pole) with decreasing frequency from a starting point of 64dB at the top of the audio frequency spectrum (20 kHz).

This means that, at the lower end of the audio frequency spectrum (20 Hz), the loop gain reaches approximately 360dB. This is egual to a loop gain, expressed as Av, of 1000,000,000,000,000,000.
That is the "loop gain" - the actual open loop gain of the amplifier, at this point, is this figure multiplied by the feedback factor, which is ~20. So at 20Hz the open loop gain is 20,000,000,000,000,000,000.

The prototype amplifier clipps at 15W average ("rms") into a 4 ohm load, which equals ~11V peak. So, at the onset of clipping at 20 Hz the virtual earth error signal will be approximately equal to 11V / 20,000,000,000,000,000,000Av  = 0.00000000000000000055V peak.

If I have my math right and haven't missed a zero here or there, that is equal to 550 zetta volts. Does anyone know if HP or Keithely ever made a zetta-voltmeter? 

LTspice doesn't even have the computational resolution to compute the loop gain as DC is approached. It bombs out even before getting down to 20Hz:

[JuKu]

In other words, noise and parasitic effects dominate. Now, those do make control theory "interesting". have you measured your prototype THD+N vs frequency and other conventional audio quality characteristics? and, btw,  !

[GK]

Well, bandwidth and rise time are determined by the input filter having a corner frequency of 159kHz. There is effectively no slew rate limiting, as the input filter is the limiting factor. I have not been successful at measuring any distortion products out of the amplifiers noise floor. However keep in mind, this initial prototype is only a low power thingie, with a closed loop gain (20 Av) relatively high compared to its power output. The input sensitivity is 350mV and the input-referred noise is approximately 8nVsqrtHz. in a 20kHz noise bandwidth that gives a S/N ratio of approximately 109dB.

8nV of input noise is pretty good for a power amplifier, but not exceptional. However I did not design the prototype for ultimate noise performance as that wasn't my main concern at the time. It would be relatively trivial though to get it down to 5nV or so, mostly by lowering the feedback impedance's and further modifying the gain distribution in the integrator stages.

5nV is I think sufficient and is what I am aiming for now, as that will give me a quite healthy S/N ratio of 126dB with an input sensitivity of 1V rms in a 20kHz noise bandwidth (some dB better A-weighted). However that is part of a new design based on my high-order control loop concept, further details of which I am rather obliged to keep under my hat



 








 

[GK]

Well it has been a few evenings now and the amplifier is still pumping out tunes here in my study. Funny enough though, after many hours of soak testing I actually did finally identify an intermittent instability, funnily enough though, only occurring when the amplifier was cold and just switched on. However it absolutely had nothing to do with the 5th control loop - forcing the amplifier into 1st order mode by overriding the logic control signal to the analog switches did not cause the oscillation to stop. The oscillation itself was a rather low level one (a very clean sine wave of about 50mV peak at the speaker output) at a fraction over 10MHz.

With the amplifier taken back into the "lab" I tracked the cause down to a relatively high Q open loop gain peaking in the integrator bank at slightly above 10MHz. I did this by breaking the global feedback loop and sweeping the amplitude response of the integrator bank with an RF signal generator and 'scope. This peaking in the open loop response turned out to contribute just enough of a degradation to the amplifiers gain margin to cause the intermittent, "on the verge" instability issue.

Fortunately though the cure was a very simple one - I just incorporated a pole into the loop by fudging in an RC filter formed by a 1K resistor and a 15pF cap to neuter the peaking with its attendant advancement in phase, to ensure instead that the loop gain continues to roll off at a deliberate and controlled rate. Exactly why the integrator bank exhibits this high frequency peaking (even in 1st order mode when the integrators are transformed into basic inverters) I'm not exactly sure at the moment, but I suspect it is due to a parasitic interaction with some characteristic of the analog switches. This is something that will require further and thorough investigation.

While this isn't exactly an "ordinary" power amplifier design, this little issue does illustrate the importance of having to characterize the amplitude and phase responses of all gain blocks within a closed negative feedback loop out to frequencies well beyond the point at which the loop gain crosses through unity.

Just thought that might be a interesting tidbit for all those control engineers out there who might be reading. 

[Tepe]

I certainly follow your R+D progress. You have at least ONE listener :-)

Yes, in fact more than one

[GK]

With the amplifier taken back into the "lab" I tracked the cause down to a relatively high Q open loop gain peaking in the integrator bank at slightly above 10MHz. I did this by breaking the global feedback loop and sweeping the amplitude response of the integrator bank with an RF signal generator and 'scope. This peaking in the open loop response turned out to contribute just enough of a degradation to the amplifiers gain margin to cause the intermittent, "on the verge" instability issue.

Found the problem. I was that silly ground plane separation I put between the analogue switches operating on the integrators and the actual integrators themselves. At the time I layed out the proto PCB I had not yet considered how I was going to later bodge on the experimental control circuitry, so I put in this split (not necessary for the purpose anyway) in an attempt to make sure the ground returns for the control circuitry and the signal circuitry are separated down to the hierarchical star grounding back to where the power enters the PCB.
I should have known better than to have split the grounds between the two intimately coupled sections of analogue circuitry. The ~10MHz peaking was caused by some kind of regeneration between the integrator stages due to feedback through the effective parasitic capacitances to ground of the analogue switches feeding into the high inductance of the effective ground loop. Bridging the ground plane split between each analogue switch/integrator combination eliminated the peaking.

[robrenz]

That's exactly what I was thinking right off the bat, but I wanted to see if you found it

[GK]

OK then, perhaps I should commission you to layout the PCB for the magazine article then?  Id be able to put my feet up for the next several evenings.
There were still a lot of unknowns when I began laying out that proto PCB and it was done with expediency as the highest priority. However as a foundation for a "proof of concept" it has served its function. Still pumping out tunes here in my study too 

[GK]

Some cross-forum pollination here - I've noticed that this thread has become a topic of conversation over here: http://www.diyaudio.com/forums/solid-state/240712-cfa-topology-audio-amplifiers-152.html#post3661711


Might as well use this as an opportunity to spruik the magazine article I mentioned previously. The magazine in question is this one: http://www.linearaudio.net/

For the basis of the article I've designed a complete and functional audio power amplifier based on the high order feedback principle developed during the course of this thread. It clips a fair bit above 100W into 4 ohms. I am currently in the final stages of testing the prototype. A rather unrevealing and small teaser picture of the amplifier under test, connected to my distortion analyser, is attached. There is just a little bit of optimization of the frequency compensation to be completed this weekend, that I wish to do just to satisfy myself that the amplifier as implemented is performing the best that it can. Then I just have to find a quiet free day (a bit difficult lately) to finish formatting the article and diagrams. To the best of my knowledge it will be published in either the upcoming issue of LA or the one after.

The amplifier developed has:

1) A fourth-order global feedback loop
2) a ULGF pushed as high a practical given stability constraints and
3) OIC in the form of TMC applied to the OPS.

There has been much heated debate about the merits of TPC versus TMC, but no one so far seems to have had the imagination to realize that the two schemes are not mutually exclusive, which is unfortunate. The amplifier developed demonstrates that this is true; though I have gone one or two steps better, of course, with "4PC" rather than just TPC.

That's about all I can say really, without spoiling the article. If the acronyms used above confuse anyone over here, I'm afraid that you'll just have to purchase the article when it finally comes out   

[Experimentonomen]

I have found that low THD amplifiers tend to be so clinical and accurate that the sound ends up a bit harch/boring and unpleasing, as the human ear kinda expects the distortion of the average amp circuit found in most consumer amplifiers which makes the sound warm and pleasing.

[GK]

, as the human ear kinda expects the distortion of the average amp circuit.

That's a good one - haven't heard that before and must add it to the list. BTW, why is it always the "Human ear"? Is that distinction necessary? Would the intended audience be otherwise unduly confused if it was just "ear"?