Different type Precision Unity Gain Follower

[mawyatt]

This all started over here but thought it better to have it's own thread.
https://www.eevblog.com/forum/testgear/ac-rms-dmm-tests/msg6190935/#msg6190935

This is about creating a low frequency Unity Gain Follower based upon an op-amp or two. The traditional method uses an op-amp with the inverting input connected to the output and has a simplified transfer function of Vo/Vi = 1/[1+1/A(f)], where A(f) is the frequency dependent op-amp gain. As |A(f)| -> ∞, the |Vo/Vi| -> 1. However op-amps have finite gain, usually very large, and rolls off with frequency to unity at the op-amp GBW Product. The finite frequency dependent gain imposes a limit on the follower accuracy as shown by the above equation. For example with |A(f)| = 100 the Unity Gain configured op-amp has a gain of 1/[1+1/100], or 0.990099, not exactly unity!!!

Utilizing an identical op-amp in the feedback (consider a dual amp package) which is configured as a direct wired unity gain follower should allow partial compensation for A(f) and produce a more accurate unity gain result (see notes). Using the same |A(f)| = 100 in this configuration produces a Unity Gain configured dual op-amp gain of (100^2+100)/[100^2+100+1], or 0.999901, within -100ppm of unity. This is an ~100X improvement toward an ideal unity gain by just adding another op-amp with no addition precision components required. Of course a little more analysis will reveal that the feedback amplifier introduces additional complex phase delay in the feedback and needs compensation for this configuration to be stable.

Attached are a couple LTspice simulation results with a 10V sine-wave at 100Hz as the input to a standard op-amp unity gain follower and dual amp version utilizing an OP-07 (LTspice model). This simulation includes a running RMS of the difference between the input and output for both follower types. Note the difference error of the dual amp version and the simulation numerical limitations hinted in the graph. Also one needs to consider the levels involved and how they relate to the op-amp Spice models, these levels are below 1ppm and should be questionable.


Anyway, this is an indication of the improvement one can achieve by just using another op-amp in the feedback of a Unity Gain Follower. Just a note, the frequency compensation is not optimal for the OP-07, it's from the LM358 version which will be shown later.
Best

[mawyatt]

Here's the results with a LM358 behavioral spice model.

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[mawyatt]

Here's an AC analysis showing the improvements possible with the Dual Op-Amp Unity Gain Follower. These are LPspice generated results based upon the difference (error) in Vout-Vin with AC Analysis (Linear). OP-07 is using same frequency compensation as LM358.

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[moffy]

Just for clarity, I am sure mawyatt knows this, but for others like me, the second graph of 'V(v01) - V(vin)' is mostly due to phase shift rather than amplitude error which should be around 50uV@100Hz, you can see it when you compare the relative phases. Not to detract, it is a real error which the configuration alleviates.

https://www.analog.com/media/en/technical-documentation/data-sheets/OP07.pdf

[mawyatt]

The LM358 and OP-07 both have ~80dB open-loop gain at 100Hz, which in a standard unity gain follower introduces ~1mV error with a 10V input without considered the effects of phase which is shown in the time domain plots. By just using Vout-Vin as the "error" this includes both amplitude errors and phase shifts, sort of a worst case.

The improvement is quite remarkable IMO considering how simple this is with general purpose op-amps. However the simulations and models are being pushing into questionable regions of dynamic range as can be seen in the "numerical noise" in the time domain plots, so some caution is advised!!

The Dual Op-Amp Unity Gain follower might prove useful around DC as well. We put together a quick plug in breadboard with both the OP-07 and LM358 versions for a sanity check and it seems to work, altho we have no way to validate to the levels of these simulations or analysis. A more detailed analysis including stability is necessary before recommending this tho, but that's something we'll leave to others, as our analysis skills were limited before and now fading fast as we become more senile

Anyway, like the Squarewave AC Reference which started all this, we thought this simple circuit was interesting and hope others will also

Added results with OP-27 using same frequency compensation as LM358 and OP-07.

Best   

[Kleinstein]

One can not ignore the phase. There may be 1 mV difference from input to output for the simple follower. However this is essentially in quadrature so the 1 mV would not directly subtract from the amplitude. So the error is smaller than it seems.

The version with the extra OP-amp in the feedback is relying on matching between the amplifiers - this work perfect in the simulation, but is limited in reality. This is especially the case when both OP-amps see a different loads at the outputs.

A more accurate alternative (though a little more effort) can be a bootstrapped (or driven from the input) supply for the buffer.

[magic]

The finite frequency dependent gain imposes a limit on the follower accuracy as shown by the above equation. For example with |A(f)| = 100 the Unity Gain configured op-amp has a gain of 1/[1+1/100], or 0.990099, not exactly unity!!!

This is a common misconception, pointed out by Kleinstein above. In reality, gain accuracy may be much better if open loop phase response at 100Hz is 90° and DC gain is higher than 100.

I ran into this problem myself in the past, if you click below and scroll down there is a small time domain simulation which demonstrates how it works.
https://www.eevblog.com/forum/projects/a-few-words-on-so-called-feedback-equation-and-closed-loop-gain-accuracy/

Curiously, it was you who pointed out that the simple equation does work correctly if complex numbers are used to account for open loop phase response

[mawyatt]

The finite frequency dependent gain imposes a limit on the follower accuracy as shown by the above equation. For example with |A(f)| = 100 the Unity Gain configured op-amp has a gain of 1/[1+1/100], or 0.990099, not exactly unity!!!

This is a common misconception, pointed out by Kleinstein above. In reality, gain accuracy may be much better if open loop phase response at 100Hz is 90° and DC gain is higher than 100.

I ran into this problem myself in the past, if you click below and scroll down there is a small time domain simulation which demonstrates how it works.
https://www.eevblog.com/forum/projects/a-few-words-on-so-called-feedback-equation-and-closed-loop-gain-accuracy/

Curiously, it was you who pointed out that the simple equation does work correctly if complex numbers are used to account for open loop phase response

Yes we know the op-amp gain is not a simple scaler (actually not a simple leaky integrator either) as we stated way back in 2022 and you pointed out.

Our analysis is much simplified, and mentioned a more detailed analysis is required for stability and such. The time domain Vout-Vin results shows the phase shift @ 100Hz (within the spice models limitations) relative to the input and the error amplitude reduction of the Dual Amp approach, the RMS computations within the simulation also reveal the improvement with this approach.

Having a higher loop gain at DC or more bandwidth as shown by the OP-27, results improve results, both in the simple and Dual Amp versions. The OP-07 and LM358 have similar loop gain @ 100Hz and simulations show similar results. The OP-07 has higher "DC" loop gain, while the LM358 has higher "leaky integrator" corner (BW) resulting in similar loop gains @ 100Hz, the OP27 has a higher "leaky integrator" (BW) corner and thus higher loop gain at 100Hz and results show this improvement.

Anyway, it's such a simple circuit which appears to have maybe some benefit.

Best 

[mawyatt]

One can not ignore the phase. There may be 1 mV difference from input to output for the simple follower. However this is essentially in quadrature so the 1 mV would not directly subtract from the amplitude. So the error is smaller than it seems.

The version with the extra OP-amp in the feedback is relying on matching between the amplifiers - this work perfect in the simulation, but is limited in reality. This is especially the case when both OP-amps see a different loads at the outputs.

A more accurate alternative (though a little more effort) can be a bootstrapped (or driven from the input) supply for the buffer.

We mentioned a more detailed analysis is required, but these results show some benefits of Dual Amp approach. We also mentioned the dual amp matching earlier in this thread and a dual op-amp should have a reasonable match between dual amps on a single die one would think. The GBW Product of the 2 op-amps on the same die should yield a reasonable "hint" at what one could expect wrt device matching.

We purposely mismatched a OP-07 version where the feedback amp had it's open-loop gain attenuated by 10% and this still shows a ~19dB improvement @ 100Hz in both Transient and AC simulations as shown.

David Hess provided a link in the AC RMS Reference thread to this old ADI AN-107 AP Note (reprint from EDN Sept 17-1987), highly recommend and note the "Analyzing Compensation Techniques"
https://www.eevblog.com/forum/testgear/ac-rms-dmm-tests/?action=dlattach;attach=2761965

Anyway, seems an interesting circuit.

Best

[mtwieg]

It's an interesting idea, one I've not encountered before. Impossible to predict matching of Aol between channels on the same die, but should be enough to make a good improvement.

However output loading might throw things off entirely, especially if the opamps have common emitter/source outputs (always the case with rail-to-rail outputs), for those Aol will be very dependent on load impedance. So both opamps should see the same load. I guess you'd want to add a dummy load to the output of the feedback opamp equal to the load seen by the forward opamp.

Wondering what the simulations look like if you use rail-to-rail output opamp model.

[mawyatt]

We have not seen this before either and the results are surprisingly good for such a simple addition!!

The ADI AN-107 Application Note indicates that many dual or quad op-amps have GBW matching to 1~2%, some are guaranteed. Here's a brief simplified analysis of having unequal op-amp gains and the effects of such on the Vout-Vin Error, and a LTspice plot of these effects on a OP-07 based Dual Amp Follower.

Output loading (unequal) has a similar effect to unequal Op-Amp GBW products, but as mentioned is easily managed in actual use. Actually one could use unequal loading as a means to tune the response, even placing a pot between the forward and feedback op-amps outputs with the wiper to ground as the tuning element.

BTW the analysis below shows a result of -97.59dB @ 100Hz for Vout-Vin and -118.52dB with the OP-07 Dual Amp Follower, which agrees with the LTspice simulation shown with a 10% and 1% OLG reduction on the feedback amp respectively mimicking an op-amp mismatch.

Anyway, these results are obviously limited by the op-amp spice models and matching but seem to "hint" this unique configuration could indeed produce a marked improvement in Unity Gain Voltage Precision Followers in spite of some naysayers

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[Kleinstein]

Using the load to compensate for different load would be tricky, as the load current depends on voltage at any time.
As a complication the GBW can be different between the cases for current sourcing and sinking.
The errors from the OP are also no just the limited gain and GBW, but there is also cross over distortion (especially with the LM358).  Chances are that a real world LM358 would behave quite different and the cross over in the extra buffer in the feedback could well cause stability issues.

[mawyatt]

It's an interesting idea, one I've not encountered before. Impossible to predict matching of Aol between channels on the same die, but should be enough to make a good improvement.

However output loading might throw things off entirely, especially if the opamps have common emitter/source outputs (always the case with rail-to-rail outputs), for those Aol will be very dependent on load impedance. So both opamps should see the same load. I guess you'd want to add a dummy load to the output of the feedback opamp equal to the load seen by the forward opamp.

Wondering what the simulations look like if you use rail-to-rail output opamp model.

Here's results with a LTC6241 CMOS RIRO Op-Amp, we reduced the supply voltages to +-5V and the input to 1Vp.

Best

[mawyatt]

Using the load to compensate for different load would be tricky, as the load current depends on voltage at any time.
As a complication the GBW can be different between the cases for current sourcing and sinking.
The errors from the OP are also no just the limited gain and GBW, but there is also cross over distortion (especially with the LM358).  Chances are that a real world LM358 would behave quite different and the cross over in the extra buffer in the feedback could well cause stability issues.

It's probably not a good idea to use the load (pot) to tweak the error response, even tho it might work for a small change to "null out" the error.

The LM358 isn't exactly what one would chose in a precision application, we just showed that the Dual Amp technique also works with this lowly GP op-amp. Also the Spice model used is behavioral and questionable for how well it models the op-amp subtle effects, and the actual physical part likely wouldn't perform as well as the simulations. However, one might expect a dual precision op-amp to perform well, the old OP-470 (quad) has 1% matching according to ADI AN107, likely others as well.

Anyway, this Dual Amp Precision Follower seems to perform quite well indeed, maybe someone has the resources (TE) to verify this technique with precise measurements, we don't!!

Best

[mawyatt]

Here's what the large signal +-10V pulse response looks like for the OP-07 based Dual Amp Follower. Note the small overshoot and undershoot of the Dual Amp Follower, indicative of the slight underdamped response.

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[smaslan]

I don't know what exactly are you trying to achieve, but if you look for precision unity gain buffer I would suggest bootstrapping. It is a bit more complex, but also just two opams and it can be pretty fast too. Even single stage bootstrapping can reach errors in order of ppm and urad at 100kHz if done properly. At 1kHz it is way below ppm and urad.

[mawyatt]

I don't know what exactly are you trying to achieve, but if you look for precision unity gain buffer I would suggest bootstrapping. It is a bit more complex, but also just two opams and it can be pretty fast too. Even single stage bootstrapping can reach errors in order of ppm and urad at 100kHz if done properly. At 1kHz it is way below ppm and urad.

Bootstrapping a unity gain op-amp follower as we understand reduces the effects of common mode but does nothing to alleviate the gain error due to finite and frequency dependent op-amp Open Loop Gain, if we are missing something (likely) please enlighten!!

The technique we've introduced (see ADI AN107) is not only very simple, requiring just another same type op-amp (best a dual for better matching) and two Rs and a C, all non-critical). This technique helps alleviate the error due to finite op-amp open loop gain and the common mode error as both op-amps "see" similar input common-modes that track the signal.

The unity gain error is reduced because the feedback unity gain configured op-amp partially compensates for the forward unity op amp gain, see the simplified analysis we provided which has been verified by simulations. This results in an output-input differential error for the standard op-amp unity gain and the dual op amp as shown in the analysis and simulation plots, which show the marked imporvement. Of course the drawback to this improvement is the bandwidth becomes limited for the improvement, but not an issue for low frequency (or DC) use.

The simplified Transfer Function for a Standard Unity Gain Op-Amp Follower is:

T(f) = A(f)/(A(f) + 1) which converges to unity as A(f) -> ∞

whereas the T(f) for the Dual Amp Unity Gain Op-Amp Follower is:

T(f) = [A(f)^2 + A(f)] / [(A(f)^2 + A(f) + 1)] which obviously converges to unity much quicker as A(f) -> ∞ because of A(f)^2 term in the numerator and denominator.

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[David Hess]

A more accurate alternative (though a little more effort) can be a bootstrapped (or driven from the input) supply for the buffer.

Bootstrapping has the advantage of not depending on the matching between the operational amplifiers.

I do not remember seeing bootstrapping used for this purpose either.  I have used it for improving common mode rejection, and trimming input bias current.

Yes we know the op-amp gain is not a simple scaler (actually not a simple leaky integrator either) as we stated way back in 2022 and you pointed out.

For those who are not familiar with this, DC and low frequency open loop gain is limited by thermal feedback from the output to input transistors, and is one of the reasons loading should be minimized on precision operational amplifiers.

Anyway, this Dual Amp Precision Follower seems to perform quite well indeed, maybe someone has the resources (TE) to verify this technique with precise measurements, we don't!!

I think the standard test for measuring open loop gain and non-linearity can be adapted, with the feedback connection replaced with a divider to the non-inverting input.

[mawyatt]

For those who are not familiar with this, DC and low frequency open loop gain is limited by thermal feedback from the output to input transistors, and is one of the reasons loading should be minimized on precision operational amplifiers.

Doubt any of the spice op-amp models include this (thermal effects)!! Interestingly IBMs SiGe bipolar transistors in BiCMOS had thermal coupling models of 2nd order which included proximity effects (required simulations post layout).

Recall an old 70s IEEE paper (believe Solomon) that showed the early 741 which had just enough thermal feedback that the OLG input phase reversed!! The inputs effectively swapped polarity, but only at very small input levels, a larger input swamped the thermal induced polarity reversal and the op-amp behaved normally!! This was before George Erdi invented the cross-coupled-quad which eliminated this and other (gradients) effects on the op-amp inputs.

Best

[smaslan]

The basic principle of bootstrapping is shown in this fig:

First stage generates floating rails for the next stage(s) which then just corrects residual error between its input and its floating common mode. Noise and distortion is mostly given by the last opamp. It can be improved in many ways like multistaging where every following stage corrects residual errors a bit more. Or closed loop generation of the floating rails. It is also possible to supply the first opamp from the floating rails too. That will effectively eliminate its input common mode admittance. Combined with total active guarding of the whole buffer it is possible to reach femto-farad input capacitances. But it is a pain to make it stable.


Practical realization is a bit more tricky. I usually go with something like this:

not optimal, just principle, but usually works fine enough for me. One big problem is simulation because about half of LTspice opamps do not implement supply rails in the model, so floating them won't appear in the transfer in any way. Kind of unpredictable.

These buffers are quite popular in metrology. One of the first appearances in metrology journals (if not the first) are from Ilya Budovsky (not a surprise, right? :-):
https://sci-hub.hlgczx.com/10.1109/cpem.1998.700047
https://sci-hub.hlgczx.com/10.1109/cpem.2010.5544183
But the idea is not new. I recall seeing those in some practical electronic journals from 70s. But back than it was used to extend voltage range of low volt opamps rather than for accuracy. Also to reduce the input capacitance for some niche audio amps.

[mawyatt]

That's an interesting and complex bootstrapping circuit, thanks for showing

These don't seem to directly compensate for finite frequency dependent op-amp OLG, other than using a slight +dBV gain which won't be accurate over frequency where the OLG begins to fall off (actually goes the wrong way!!) and won't track the variation in OLG with supply voltage, temperature, aging and so on, where the op-amp feedback approach will partially compensate these effects within the matching and tracking within a dual op-amp package. David Hess mentioned the ADI AN107 which is the only source we've seen, since just recently discovering this Dual Amp Unity Gain Follower topology ourselves, that utilizes a similar op-amp in the feedback loop for compensation, but it doesn't reference unity gain use.

BTW we can't view the links above. Also our interpretation of audio is that it's all about distortion not precision, a 10ppm change in amplifier gain isn't an issue while a 10ppm change in distortion could be an issue.

Best

[Kleinstein]

The bootstrapping should also help with limited gain. The floating OP-amp that does the final drive / decision only sees the much reduced amplitude. So there is only the fraction of a much smaller voltage.
In practive one may have to have different speeds for the 2 amplifier parts - often a lower BW for the driven supply.
So there is often something like a factor of 10 less than the gain squared. At relatively low frequency this many still be better than the OP-amps gain times the matching accuracy (e.g. a factor of 10 to maybe 100 if lucky).

I have found a link to the use of a boostrapped buffer for AC. They do it more for high input impedance, but they also see an improved phase shift, which is the main issue with the limited gain / GBW.
 https://bibliotekanauki.pl/articles/114359.pdf

[smaslan]

Yes, phase error is improved drastically with this topology. It can drop to few urads at 100kHz. Input impedance is kind of a problem. You can have buffer with below 100ppm and 100urad error at 1MHz, but input impedance is very frequency dependent and when used e.g. as output buffer for resistive voltage divider, it will make bigger error than buffer transfer itself. But it is very hard to maintain high input Z at high freq even with full guarding. Years ago I made one and it was quite a pain to make it at least conditionally stable for certain input impedance:
https://github.com/smaslan/QuADC-buffer

The two older articles from Ilya are:
Budovsky, I.; Gibbes, A.M.; Arthur, D.C. . (1998).  [IEEE Conference on Precision Electromagnetic: Measurements - Washington, DC, USA (6-10 July 1998)] 1998 Conference on Precision Electromagnetic Measurements Digest (Cat. No.98CH36254) - A high-frequency thermal power comparator. , (), 544–545. doi:10.1109/cpem.1998.700047
Budovsky, I.; Hagen, T. . (2010).  [IEEE 2010 Conference on Precision Electromagnetic Measurements (CPEM 2010) - Daejeon, Korea (South) (2010.06.13-2010.06.18)] CPEM 2010 - A precision buffer amplifier for low-frequency metrology applications. , (), 28–29. doi:10.1109/cpem.2010.5544183 
Both are behind paywall (but accessible via SciHub).

[mawyatt]

The Dual Amp Unity Gain follower idea came from the discussions revolving around creating a precision low frequency AC Reference based upon a reference DAC and unity gain buffer amp discussed over here:

https://www.eevblog.com/forum/testgear/ac-rms-dmm-tests/msg6185139/#msg6185139


In this AC Reference application phase/group delay is meaningless. If one is interested in phase response then a look at ADI App Note AN107 is in order. The Dual Amp unity gain follower we've introduced corrects for finite frequency dependent gain, improves low frequency Common Mode and Power Supply rejection, all from an extremely simple concept of utilizing the same type (dual op-amp package, example OP227) op-amp in the feedback of the unity gain configured follower.

Since the characteristics of the dual op-amps are quite similar on a single die/package (see AN107), the behavior of negative feedback "inverting" the feedback function tends to compensate for the forward op-amp undesirable characteristics such as finite frequency dependent gain and such. This is because these characteristics "appear" in the feedback path due to the feedback op-amp and partially cancel the forward op-amp characteristics due to the nature of negative feedback mentioned.

Caution is advised as this is all theoretical and only verified by simulations which are limited by the op-amp models and spice dynamic range. We don't have the means to evaluate at these levels and will leave this for others to discover.

Anyway, don't think one can get much simpler than this approach and achieve similar improvements, at least at the lower frequency range. Here's some LTspice results using the Dual Amp OP227 which has guaranteed matching between op-amps. These plots show Output-Input Error and Power Supply Rejection improvements by the Dual Amp unity gain follower from a Standard unity gain follower.

As always YMMV

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[Kleinstein]

For looking at the amplifier it is not enough to look at the difference from input to output. This is especially the case if not taking the phase into account. The simple buffer has a phase shift of some 90 degree between the signal and the error. For this reason the error is mainly a phase shift and only very little amplitude error.

For the simulation one can zoom in and also directly look at the amplitudes. Here it depends on the symmetry if the AN107 type buffer is really much better than the simple buffer in the amplitude. The AN107 type buffer still has a much smaller phase shift. Only with good matching the extra OP-amp in the feedback also helps with the amplitude.

Even the simple buffer is not that bad. In many cases a simple buffer with a fast, low THD OP-amp (e.g. OPA1655) would be good enough  (low ppm error range) for the lower frequencies, like up to 10 kHz.

I have done a simulation with the AD820 model, that also supports a bootstrapped supply. Because of the rail to rail output stage this OP-amp may be rather sensitive to different loading. Matching may get a little better with something like the OP227. The advantage of the AN107 like buffer is more with the phase. The simple bootstrapped version also does not fair well - it however gets better than an additional in loop buffer for the output, that would help reduce the loading effect (and thermal effects).