Different type Precision Unity Gain Follower

[mawyatt]

Here's result showing the Power Supply Rejection (PSR) using a SDS824X HD Bode Plot function (amazed it can even show this type of measurement!!). The setup was a plugin (white) board with 3 random OP07s from our scrap parts bin. These OP07 are questionable as can't remember when/where they were purchased (could be counterfeit) and how much "abuse" they've seen!!

So basically this is about as bad a setup with unknown random parts as you can get!! One of the op-amps was configured as a unity gain follower and the other two were configured as a Dual Amp follower as shown previously. The AWG was set to supply Vcc at +7.5VDC with a 5VPP sinewave, while the Vee was supplied with a lab supply at -7.5VDC. The DSO and Bode function was set to 20MHz BW on all channels and 10X probe used for CH1 (Ref) and 1X probes for CH2 (Follower) and CH3 (Dual Amp Follower).

As you can see from the actual plot the Dual Amp and Standard Follower start off at 10Hz with the ~same PSR attenuation which hints that the op-amps in the Dual Amp are not even close to being matched like they would in a dual amp die/package. Yet these random OP07s produce a ~21dB improvement in PSR at 100Hz

Best

[Kleinstein]

The large dyncmic range for the bode plot is indeed impressive. There is not that much signal left at 90 dB PSRR.

[mawyatt]

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).

Here's a simulation with the AD820 and results shown as Vout-1 since the input is defined as a magnitude of 1 and phase zero (makes no difference if the results is Vout-Vin or Vout -1 in this case). -86.36dBv at 100Hz is ~48ppm error, while -143.97dBv is ~63ppb.

Also included a time domain plot thats shows the time response of the Standard and Dual Amp approach, note the scales

In addition we've included the running RMS error results of Vout-Vin directly from LTspice, note the DRAMATIC IMPROVEMENT IN ALL CASES (FREQUENCY DOMAIN, TIME DOMAIN, RMS ERROR COMPUTATIONS) THE DUAL AMP APPROACH PRODUCES

Please spend some time studying the simplified analysis we provided much earlier, maybe even expand it with a complex Oprn-Loop-Gain analysis, and the various simulations produced with different op-amps, frequency and time domain, running RMS waveform calculations, and on and on!!

This technique also improves PSR with no additional effort or components as previously shown.

Best

[mawyatt]

The large dyncmic range for the bode plot is indeed impressive. There is not that much signal left at 90 dB PSRR.

Yes, these little Siglent SDS824X HD are remarkably good, and the Bode is icing on the cake

Can't say enough good things about this little guy

Best

[Kleinstein]

Here's a simulation with the AD820 and results shown as Vout-1 since the input is defined as a magnitude of 1 and phase zero (makes no difference if the results is Vout-Vin or Vout -1 in this case). -86.36dBv at 100Hz is ~48ppm error, while -143.97dBv is ~63ppb.

I get the same ~ 48 ppm for the difference from input to output. However the 48 ppm are at near 90 degree ( I got some 90.4 deg with 100 K load). When adding as vector (or complex number) this only give a change in the amplitude of some -0.37 ppm. Most of the effect comes from the small deviation from 90 deg. and this way got the slightly smaller amplitude. The point is that AC voltages add as vector.
The Buffer following AN107 still performs a little better with -0.29 ppm and only some 300 nV difference, but this time the difference is nearly in phase (171 deg).

The surprising point is that the simple buffer is better than one might think. Much of the error is only a phase shift.

[mawyatt]

Here's a simulation with the AD820 and results shown as Vout-1 since the input is defined as a magnitude of 1 and phase zero (makes no difference if the results is Vout-Vin or Vout -1 in this case). -86.36dBv at 100Hz is ~48ppm error, while -143.97dBv is ~63ppb.

I get the same ~ 48 ppm for the difference from input to output. However the 48 ppm are at near 90 degree ( I got some 90.4 deg with 100 K load). When adding as vector (or complex number) this only give a change in the amplitude of some -0.37 ppm. Most of the effect comes from the small deviation from 90 deg. and this way got the slightly smaller amplitude. The point is that AC voltages add as vector.
The Buffer following AN107 still performs a little better with -0.29 ppm and only some 300 nV difference, but this time the difference is nearly in phase (171 deg).

The surprising point is that the simple buffer is better than one might think. Much of the error is only a phase shift.

If you believe that the unity gain error is all due to phase shift and not the available gain at frequency A(f), then do a simple analysis of an ideal (no frequency dependence nor phase shift) unity gain with op-amp of ideal Open-Loop-Gain A, where A >>1 and you find the error between exactly 1 is ~ 1/A. Thus an ideal op-amp of OLG of 80dBV in unity gain configuration has a gain error of ~ -80dB. Since this is ideal with no frequency dependance, there's no additional phase shift involved!! You can create an ideal op-amp in spice with either a VCVS (E) or if you what to include the frequency dependance with phase effects a VCCS with a shunt RC, where RC sets the leaky integrator frequency corner.

Here's an example of an ideal no phase shift, no frequency dependance, op-amp (VCVS) in a unity gain configuration in Green with a OLG of 8000 (78dBV). A VCCS based op-amp in unity gain configuration with an OLG of 800,000 (118dBV) and leaky integrator corner of 1Hz (to mimic a OP07) in Red, and a OP07 in Deep Blue. Note they all exhibit almost exactly the Vout-1 (or Vout-Vin if you prefer) error at 100Hz as -78dB wether they have a phase shift core op-amp or an ideal no phase shift op-amp follower configuration, or OP07 model. Now look at the OP07 based Dual Amp follower in light blue and the significant improvement over any of the standard unity gain configurations, wether created with ideal frequency independent no phase shift based op-amp, simple VCCS model op-amp or spice model OP07.

Evidently the phase shift doesn't matter as the no phase shift ideal based op-amp VCVS produces the same result as the VCCS with phase created by leaky integrator model, and with the full OP07 spice model. 

Anyway, hope this helps illustrate where we are coming from regarding this Dual Amp Follower and it's significant benefits.

Best

[Kleinstein]

The no phase shift case only applies to really low frequencies, like < 0.3 Hz for the OP07. The
VCCS model voltage follower mimics the OP07, including the phase shift.
It is only the artificial VCVS model that has no phase shift in the input to output differential. There the input to output difference directly gives the amplitude error.

The point with low phase shift could be relevant with older high speed amplifiers like AD826, that have a relatively low DC open loop gain. It would still be tricky to use the AN107 like circuit with such high GBW parts.

The imporance of the phase gets obvious if one looks at it the other way around:
If there is no amplitude error, but a (small) phase shift, one still gets a difference between the input and output. It than comes with the near 90 deg. phase shift for the input to output difference. So one can have a difference from input to output even without any amplitude error. If there is a near 90 deg phase in the difference it is mainly a phase error, not so much an amplitude error.

LTspice allows to directly look at the amplitude and it shows the rather small error also for the OP07 at 100 Hz. For the OP07 model follower I get 117 ppm of input to ouput difference at -90.65 deg phase. Looking directly at the amplitude (linear scale helps), the amplitude is 1.33 ppm down, with a -6.7 deg phase shift.
So the simple follower is surprisingly good when it comes to the amplitude. Much of the error is a phase shift.

The extra OP-amp in the feedback can help, but the advantage is less than simply looking at the input to output difference suggests. Much of the improvement is in the phase and at the very low frequencies. Amplitude wise the simple buffer is surprisingly good - much better than the simple scalar calculation would suggest.
The simple bootstrapped supply is similar: it helps with the phase and at very low frequencies, but little (to not at all) with the amplitude at high frequencies.

[magic]

Amplitude wise the simple buffer is surprisingly good - much better than the simple scalar calculation would suggest.

As mawyatt pointed out elsewhere, the simple calculation works if you use complex numbers to represent phase. For "ideal" 90° phase opamp it seems that gain error is 1 over loop gain squared, as long as that doesn't exceed DC loop gain.

The simple bootstrapped supply is similar: it helps with the phase and at very low frequencies, but little (to not at all) with the amplitude at high frequencies.

Is this conclusion based on simulations?

I find it dubious - if we imagine ideal bootstrap (unity gain and zero phase) then the boostrapped opamp has nothing to do (except driving output load) and should show improvement. Maybe your bootstrap was too slow or the model included some hidden references to ground.

[Kleinstein]

The simple bootstrapped supply is similar: it helps with the phase and at very low frequencies, but little (to not at all) with the amplitude at high frequencies.

Is this conclusion based on simulations?

I find it dubious - if we imagine ideal bootstrap (unity gain and zero phase) then the boostrapped opamp has nothing to do (except driving output load) and should show improvement. Maybe your bootstrap was too slow or the model included some hidden references to ground.

This from the simulation. I checked the OP-amp model. The AD820 model did not show an effect from the ground moving around. Many other models do show such hidden ground reference.

With a slightly deeper look it gets reasonable: the bootstrapped supply is off by mainly a phase shift from a simple buffer providing the Bootstrapping. The bootstrapped OP-amp has to correct for this error and it does it by leaving an error behind that is again 90 deg phase shifted. So the overall error is 180 deg. phase shifted and thus mainly an amplitude error from 2 x the OP-amp gain, and relatively low error in the 90 deg. off direction to cause phase shifts.
The article  https://bibliotekanauki.pl/articles/114359.pdf , that I had linked before also found this in a real world measurement: little help with the amplitude, but an improved phase.

I used the BS supply from the output and for stability reasones slowed down. One could expect an improvement with a fast supply driver from the input - here the fast one could get away with higher noise, less precision.

p.s. A big part of the error for the BS supply version is from the loading effect. The BS OP-amp still sees the full load and the AD820 with the RR output is very susceptible to loading. An extra fast buffer in the loop can really improve the result in the simulation. The extra buffer also makes sense to remove thermal effects from the critical OP-amp and improve the tolerance to capacitive load.

[mawyatt]

The no phase shift case only applies to really low frequencies, like < 0.3 Hz for the OP07. The
VCCS model voltage follower mimics the OP07, including the phase shift.
It is only the artificial VCVS model that has no phase shift in the input to output differential. There the input to output difference directly gives the amplitude error.

The point with low phase shift could be relevant with older high speed amplifiers like AD826, that have a relatively low DC open loop gain. It would still be tricky to use the AN107 like circuit with such high GBW parts.

The imporance of the phase gets obvious if one looks at it the other way around:
If there is no amplitude error, but a (small) phase shift, one still gets a difference between the input and output. It than comes with the near 90 deg. phase shift for the input to output difference. So one can have a difference from input to output even without any amplitude error. If there is a near 90 deg phase in the difference it is mainly a phase error, not so much an amplitude error.

LTspice allows to directly look at the amplitude and it shows the rather small error also for the OP07 at 100 Hz. For the OP07 model follower I get 117 ppm of input to ouput difference at -90.65 deg phase. Looking directly at the amplitude (linear scale helps), the amplitude is 1.33 ppm down, with a -6.7 deg phase shift.
So the simple follower is surprisingly good when it comes to the amplitude. Much of the error is a phase shift.

The extra OP-amp in the feedback can help, but the advantage is less than simply looking at the input to output difference suggests. Much of the improvement is in the phase and at the very low frequencies. Amplitude wise the simple buffer is surprisingly good - much better than the simple scalar calculation would suggest.
The simple bootstrapped supply is similar: it helps with the phase and at very low frequencies, but little (to not at all) with the amplitude at high frequencies.

Think you are totally missing the point. The op-amp phase shift isn't significantly effecting the Vout-Vin voltage peak to peak or RMS magnitude error, it's the op-amp gain magnitude at frequency that matters most. Here's time domain simulation at 100Hz showing the actual Vout-Vin error of the OP07 based Follower (Blue), the Dual Amp OP07 Follower (Red) {note scale!!}, the Ideal VCVS with 8k gain to match the 8k gain of the OP07 at 100Hz (this has no additional phase shift), and Leaky Integrator VCCS with 800k OLG to match the OP07 (this has 90 degrees additional phase shift due to the integrator). We've also included the time domain running RMS Vout-Vin error for each of these "Unity Gain Followers".

As you can see, the peak to peak or RMS Vout-Vin magnitude error doesn't change with the op-amp phase shift, wether an OP07 or VCCS with phase shift, or no phase shift VCVS, it's the op-amp loop gain at frequency that matters. The phase introduced by the op-amp (or equivalent) has little if any effect on the Vout-Vin magnitude error. The Dual Amp approach completely squashes this Vout-Vin magnitude error to sub ppm levels with just a pair of OP07s. With dual op-amps like those previously mentioned the matching should be good and results approaching this improvement could be expected.

The running RMS error LTspice computations show the Vout-Vin RMS errors are almost identical between the OP07, VCVS (no Phase shift) and VCCS (Phase shift) Followers. Note the 745X improvement with the Dual Amp Follower Vout-Vin Error magnitude which agrees with the AC analysis of -135.88dB for the Dual Amp and -78.36dB for the OP07 Follower, or 751X improvement with the Dual Amp Follower

BTW if you look at the AC Vout-Vin Error simulations, this predicts a Vout-Vin magnitude error of -78.36dB for the OP07 Follower, -78dB for the VCVS and VCCS Followers, and -135.88dB for the Dual Amp Follower. This agrees with the Time Domain simulations, for example the Dual Amp the result is 10V peak sinewave *(10^(-135.88/20)), or ~1.61uV peak, and the OP07 Follower 10V*(10^(-78.36/20)), or ~1.21mV peak. 

Don't know what else we can do to show the benefits of this Dual Amp follower wrt to unity gain precision at low frequencies, and how the standard op-amp unity gain follower, bootstrapped or not, is limited by available op-amp loop gain at frequency. Analysis shows this, AC Linear simulation plots show this, Time Domain simulation plots show this, and TD running RMS computations show this

Best

[Kleinstein]

The point is that the difference between the input and output is not the same as the gain error. It combines the phase shift and amplitude error.

With the simple OP-amp and not very low frequencies (e.g. 100 Hz) the difference between the input and output is essentially out of phase from the main signal.  Adding such an out of phase voltage to the input gives mainly a phase shift, but little amplitude error. The simulation also shows the small amplitude error, if one directly looks at the amplitude.

So the simple buffer is already much better in the amplitude than just looking at the input to output difference suggests. Most of the input to output difference is from the phase shift.

[mawyatt]

The point is that the difference between the input and output is not the same as the gain error. It combines the phase shift and amplitude error.

Did you bother to actually look at the various plots and previous analysis?? The |Vout-Vin| error is the same regardless of op-amp phase!! This is verified in the AC Linear Plots, Time Domain Plots and running RMS computations, the |Vout-Vin| Error is almost independent of the op-amp phase. Must we remind that Unity Gain means Unity Gain, that's exactly 1 and nothing else!! The Unity Gain Error is the difference between the output and unity with an input of 1, and since the input in AC analysis is Unity it can also be expressed as Unity Gain Error = Vout - 1, or Vout - Vin since Vin is 1 for AC analysis.

With the simple OP-amp and not very low frequencies (e.g. 100 Hz) the difference between the input and output is essentially out of phase from the main signal.  Adding such an out of phase voltage to the input gives mainly a phase shift, but little amplitude error. The simulation also shows the small amplitude error, if one directly looks at the amplitude.

So the simple buffer is already much better in the amplitude than just looking at the input to output difference suggests. Most of the input to output difference is from the phase shift.

The simple buffer is NOT much better in magnitude error and has been shown by simple analysis to suffer from finite op-amp loop gain at frequency. Look at the Error plots in both Linear AC analysis, Time Domain and running RMS computational results, these all agree with the analysis we provided.

The simple answer, if you chose to accept it, is that the op-amp gain magnitude at frequency limits the Unity Gain Follower Precision, regardless of op-amp phase, and Unity Gain is not anywhere near as good as you portray. We've shown this with analysis, and multiple simulations in AC, Time Domain and RMS computations.

Do you agree that the Unity Gain Follower behaves as Gain Transfer Function = A(f)/((A(f)+1), where A(f) is the op-amp gain at frequency f, or do you have another better model? Do you agree that the error in a Unity Gain Follower is anything that deviates away from a gain of exactly 1? How do you explain the simulation results we've provided which all agree with the previous analysis we also provided??

Look at the OP07 Follower, it has a A(f) of 78.6dB at 100Hz and in the Follower shows a Vout-Vin Magnitude Error of -78.36dB, that's 120ppm, which agrees with the Time Domain sinewave error of 1.2mV/10V (Vin is 10V peak, or 120ppm)!!

This is a similar result for the ideal VCVS Follower which has A(f) of 78dB everywhere with no phase shift and a Vout-Vin error of -78dB (125ppm), the Time Domain simulation also shows ~1.2mV/10V, or ~120ppm for the VCVS Follower. Can you explain this result with a op-amp VCVS model that has no inherent phase shift anywhere, yet exhibits almost identical magnitude behavior to the OP07 Follower? If you examine the Time Domain result carefully, you will discover that the VCVS Follower has a voltage minimum of -1.2mV exactly at the Input Sinewave peak 10V (120ppm) which indicates that the loop gain isn't high enough to drive that towards zero. Just for fun we increased the VCVS gain to 120dB and the result was exactly as expected a minimum of 10uV at the input 10V peak (1ppm), the LTspice simulation is cleaner since the VCVS is ideal and the "numerical noise" less.

The Dual Amp approach alleviates the dependance upon op-amp gain at lower frequencies and provides a much closer Unity Gain response. The Dual Amp Follower Transfer Function is (A(f)^2 + A(f))/(A(f)^2 + A(f) + 1) for matched op-amps, and (A(f)B(f) + A(f))/(A(f)B(f) + B(f) + 1) where B(f) is the feedback op-amp gain at frequency. It should be clear to anyone that these produce a much better result than the Standard Unity Gain Follower above, mainly due to the A(f)^2 or A(f)B(f) terms in the numerator and denominator. Again simulation results with Linear AC, Time Domain and RMS computations prove this out!!!

Best

[Kleinstein]

|Vout-Vin| is only the error of the unity gain follower if the target is perfect amplitude and perfect phase. If one looks at the magnitude and phase error separate one has to looks at  Vout / Vin  as magnitude and phase.
As the extreme case an inverter still has an magnituide gain of 1, but 180 degree phase. |Vout-Vin| / Vin would be 2, but |Vout/Vin| is still at a perfect 1. So zero magnitude error, but with a 180 deg. phase.

The disagreement is not in how the buffers behave, but what we call magnitude error, when looking at magnitude and phase separate.

For the OP07 as follower at 100 Hz there seem to be some 120 ppm of difference between Vout and Vin. For Vout/Vin Ltspice gives 0.99999866 and phase of -6.8 degrees. That is some 1.34 ppm of magnitude error and the -6.8 degree as phase error.
So if one does not care about the phase (like for measuring the RMS value) the simple buffer is already pretty good.

[mawyatt]

For the OP07 as follower at 100 Hz there seem to be some 120 ppm of difference between Vout and Vin. For Vout/Vin Ltspice gives 0.99999866 and phase of -6.8 degrees. That is some 1.34 ppm of magnitude error and the -6.8 degree as phase error.
So if one does not care about the phase (like for measuring the RMS value) the simple buffer is already pretty good.

Agree the OP07 has 120ppm Vout-Vin error and you say it has a 1.34ppm magnitude error with -6.8 degrees phase error. So explain based upon the T(f) = A(f)/(A(f) + 1) Unity Gain Follower how this 1.34ppm magnitude error is accomplished with an OP07 A(f) at 100Hz gain magnitude of 78.6dB!!!! How is the ideal VCVS Follower with zero phase shift (Vout-Vin phase is identically zero) producing a similar 120ppm Vout-Vin error.

Also please explain how the LTspice running RMS results shown for the OP07 based Follower (with phase shift), the VCVS Follower (no phase shift) and the VCCS (leaky integrator with phase shift) which all show almost identical RMS Vout-Vin error of 829, 884, and 883uVRMS respectively for a 10V (7VRMS), so 83, 88 and 88ppm respectively.

As we stated at the beginning of this thread this was created out of the idea of utilizing a precision DAC as a precision AC Reference source, possibly capable of 10ppm levels of performance for use with measurement test equipment (DMMs and such).

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

We just did a quick measurement with a KS34465A of a random OP07 (mentioned earlier these are questionable) configured as a Unity Gain Follower, driven with a 10V Sinewave (7VRMS). The measured Vout-Vin is ~977uVRMS which equates to -77.2dB and a ~138ppm Vout-Vin Error. We also measured a Dual Amp Follower with 2 random OP07s with same sinewave input, and Vout-Vin is ~150uVRMS, which equates to -93.5dB and a ~21ppm Vout-Vin Error. Obviously these random op-amps are not matched but achieve a reasonable result, would expect improved results with a good dual precision op-amp.

Anyway, we put in enough on this already. Various analysis, multiple simulations, and now a few measurements which confirm results for us, hopefully others will find this Dual Amp Precision Follower useful

Best

[gf]

So explain based upon the T(f) = A(f)/(A(f) + 1) Unity Gain Follower how this 1.34ppm magnitude error is accomplished with an OP07 A(f) at 100Hz gain magnitude of 78.6dB!!!!

The key is a complex A(f). If arg(A(f)) is close to 90° or -90°, then abs(T(f)) can be very close to 1.

Example for ideal 90°:

Code: [Select]

A=-8511i    % 78.6dB
abs(A/(A+1)) - 1
ans =   -6.9025e-09

EDIT: 1.34 ppm is achieved with arg(A(f)) of approx. 89.35°.
At 100 Hz, it is plausible that the OL phase of the OP07 could be around this order of magnitude.
However, the -8.6° (closed loop) in Kleinstein's simulation seem to be excessive, according to T(f) = A(f) / (A(f) + 1).

[Kleinstein]

When looking at the gain and phase error separate, it is about looking at the output amplitude and phase and not about looking at the difference between the input an output. The input to output difference is just convenient to measure for small errors - but that is more a work around as the instrument accuracy is limited.
So look at |Vout| / |Vin| - 1  and not |Vout-Vin|. It is the same when one looks for the -3 dB point of an amplifier. For the simulation result one gets the direct amplitude reading via the cursor.

With the phase shift one can have 120 ppm for |Vout-Vin| and still only 1.3 ppm for |Vout/Vin| -1.
The error is a bit small for a direct measurement at 100 Hz. One could measure the effect at some 10 kHz (10 V amplitude may however run in a slew rate limit). Here the in to out difference should be at some 12 mV (with 1 V in), but the direct output amplitude should still be close to 100% (some 41 ppm or 0.00035 dB up in the simulation). One may not be able to resolve the actual amplitude error, but it should be clear that the gain error is not 1.2 %.

The T(f) = A(f)/(A(f) + 1) = 1 / (1 + 1/A(f)) formula is OK, but one has to include the phase in A(f). With most OP-amps A(f) is nearly purely imaginary over quite some frequency range. This way |(1 + 1/A(f))| is a lot closer to 1 than in the case without the phase shift. One is closer to (1 + 1/|A(f)|²) than (1 + 1/|A(f)|).

[gf]

However, the -8.6° (closed loop) in Kleinstein's simulation seem to be excessive, according to T(f) = A(f) / (A(f) + 1).

Starting with T = 0.99999866 * exp(-6.8i*pi/180) = 0.99296 - 0.11840i and solving for A, I get A = -T/(T-1) = -0.49990 - 8.41596i, which is not plausible, because abs(A) is much lower than 78dB; and real(A) is negative, which I find unlikely too, at 100 Hz (100 Hz is llikely still within the 1st order roll-off region of A(f)).

OTOH, a plausible A of 78.6dB that leads to 1.34 ppm has angle(A) = 89.35°, and that would lead to an angle(T) very close to zero, not 6.8°.

Where is the mistake (or my mistake)?

[mawyatt]

There's no way at 100Hz to have that much phase shift, seems off by orders of magnitude!!

We can't measure anything to these levels (don't have proper TE) except somewhat differential Vout-Vin magnitude (actually RMS) which is what we did and why we referenced differential Vout-Vin magnitude as the Follower Error. If we assume the complex op-amp OLG is A(o)/(1+j(f/fc)), where A(o) is the DC OLG and fc is the integrator corner, the result we get for a Unity Gain configured op-amp is:

Vout-Vin = Vin[(A(o)^2+(f/fc)^2)/(A(o)^2+2(f/fc)^2+j(f/fc)(A(o)+1))]

Caution: Please check the math, we're not that good at solving things anymore 

Altho seems correct as at f=0, when Vout-Vin = Vin(A(o)^2/(A(o)^2+1)).

Anyway, we're not going to attempt the Dual Amp, will leave that for others!!

Best

[Kleinstein]

OTOH, a plausible A of 78.6dB that leads to 1.34 ppm has angle(A) = 89.35°, and that would lead to an angle(T) very close to zero, not 6.8°.

Where is the mistake (or my mistake)?

I found the mistake: the simulation give 6.8 m° , so indeed a rather small phase shift. Still the error is mainly a phase error with Vin-Vout at close to 90 deg phase.

[gf]

OTOH, a plausible A of 78.6dB that leads to 1.34 ppm has angle(A) = 89.35°, and that would lead to an angle(T) very close to zero, not 6.8°.

Where is the mistake (or my mistake)?

I found the mistake: the simulation give 6.8 m° , so indeed a rather small phase shift.

That's consistent now 

Still the error is mainly a phase error with Vin-Vout at close to 90 deg phase.

Sure, while abs(T)-1 is -1.34 ppm, abs(T-1) is 119 ppm.

[mawyatt]

Here's a test before we move on to another effort that's brought on an interesting measurement result. Setup is a OP07 Follower and Dual Amp Follower with OP07, same as before. Input is a 20VPP Sinewave at 100Hz from AWG and various DMMs used to measure Vout -Vin for the OP07 Follower and the Dual Amp Follower. We used three different Keysight 34465As, DMM6500, DAQ6510 and SDM3065X. A coaxial cable was used for the DMMs input with the shield connected to the AWG output at the circuit input, the center lead connected to the circuit output. As mentioned the circuit is on a plug in breadboard that sits on a grounded metal plate to earth.

The 3 KS34465As all read zero for Dual Amp Vout-Vin as shown with at least 500 samples. We changed every AC related KS34465A measurement parameter and even changed the input leads to just two clip leads and even used another AWG (SDG6022X), all the KS readings return to zero. The circuit is working and confirmed by a DSO, input is 20VPP at 100Hz, output is 20VPP at 100Hz with no visible waveform artifacts. If we measure something else the Keysight's behave as expected!!
See TE thread:
https://www.eevblog.com/forum/testgear/keysight-34465a-cheating/


The OP07 Follower produces results shown as expected. We know the Dual Amp Follower is good, but not as good as Keysight says!!

Since the input is 20VPP (~7VRMS) these results can be scaled to ppm by dividing by 7.

Anyway this isn't about the TE (we'll start a thread over on TE for this)  but does show that the Dual Amp Follower can produce some very low Vout-Vin Errors, even with random OP07s  

Here's the Dual Amp results.

Apparently the mentioned Keysight behavior is normal (below 0.3% range) and discussed over here in the TE thread:
https://www.eevblog.com/forum/testgear/keysight-34465a-cheating/

Best

[mawyatt]

And the OP07 Unity Gain Follower.

Best

[RoGeorge]

Glad to see you are investigating the results in practice, too, not only in LTspice.

I wanted to give that a try, too, (I mean, to try your idea from this reply:  https://www.eevblog.com/forum/testgear/ac-rms-dmm-tests/msg6189579/#msg6189579 ) but I've realized I don't have any instrument to measure with enough resolution.

All I've tried so far was in LTspice.  There are many LTspice settings to get higher than default numerical precision.  I don't know them very well.  From the settings I've used before to increase the precision, I've changed the ones listed as .option in the attached .asc zipped file, and got a very different results for a simple OP07 repeater:  -26.7uV difference in Vpp between output and input.  Got these results, copied from the LTspice .log file:

vin_meas: PP(v(in))=19.9999999962 FROM 0 TO 0.05
vdiff_calc: vout_meas - vin_meas=-2.67373843243e-005     <---    Vpp_out is only 27uV smaller than Vpp_in
vout_meas: PP(v(out))=19.9999732588 FROM 0 TO 0.05

I've used Vpp instead of Vrms (in the hope that using Vpp would be easier and produce more accurate .meas results, but I didn't investigate if there is any benefit in using Vpp instead of Vrms).

Writing all these because (at a first look, without carefully reading all the posts in this thread) it looks like your single OP07 repeater produced more than 1mV RMS of error between in and out, while with the altered .option I've used, the calculated error seems much smaller, less than 27uV PP.

Maybe a large portion of that mV range RMS error (in the single OP07 voltage follower) come from the numerical rounding, results compression, and who knows what other tricks and shortcuts LTspice might be using in its defaults.  Attaching the zip with the high resolution settings I've used, the .asc and its .log file results when enforcing max look ahead to 10ns vs 100ns, run in LTspice XVII.

In case the LTspice precision was already dealt with, or not an issue, it's only my fault for not properly reading what was already posted (I'm in a dizzy situation at the moment, couldn't spend enough time, hope to catch up later).

Looking further for your experimental results/measurements. 

[mawyatt]

As stated we don't have the "proper" TE to evaluate these Unity Gain Followers and thus are just measuring Vout-Vin with a DMM using a 20VPP 100Hz Sinewave input. This measurement with the Keysight 34465A created a lot of discussions over on the TE linked thread we created to discuss the unusual zero readings of the KS DMM. Apparently the Dual Amp Precision Follower is so good that the error in Vout-Vin is below the KS DMM threshold on the 100mv AC scale that the DMM (all 3) display identically zero!! This is even with the two OP07s randomly selected from our scrap parts bin!! Of course with a proper Dual Op-Amp (better matching) this result should be improved, but the zero Vout-Vin differential as the KS DMM indicates and this created some interesting discussions.

The OP07 model likely has more influence on the simulation results, and we aren't privileged to that level of OP07 model detail. We know the LM358 model we have/use is totally behavioral, so use should be with caution at extremes. Tweaking LTspice engine parameters is a good path if one knows what the parameters influence and how this is applied in actual simulations.

In AC Analysis it's usually the numerical precision as everything is "linearized", however in Transient Analysis it's about the convergence criterion (Charge, Voltage, Current) and the Look-Ahead Time Stepping Algorithm. This can even get involved with the Time Step increments, and the exact Time Positions. Way back many decades ago with Cadence we were working with extremely high dynamic range circuits, so high that the quadratic time step interpolations were limiting the performance when doing FFTs. This was solved by modifying Spice such that the Time Steps were forced to land on exactly the required FFT data input time points, and not interpolated. The improvement in Time Domain simulation FFT dynamic range was dramatic.

Anyway, this Dual Amp Precision Unity Gain Follower actual performance, even with unmatched OP07s (dual op-amp would be better), seems to be quite good and somewhat confirming the simulations. By using the same op-amp as a unity follower in the negative feedback of the forward unity follower, some of the undesirable op-amp artifacts get suppressed. It's so simple surprised this hasn't been done before, probably has, altho we haven't seen anything as simple as this.

Best

[gf]

Writing all these because (at a first look, without carefully reading all the posts in this thread) it looks like your single OP07 repeater produced more than 1mV RMS of error between in and out, while with the altered .option I've used, the calculated error seems much smaller, less than 27uV PP.

For a a simple OP07 follower with 10V input @ 100 Hz,

|Vout(f)| - |Vin(f)|    is in the order of microvolts, and
| Vout(f) - Vin(f) |    is > 1 mV.

That's not contradictory.

The latter is what you would measure with a voltmeter between Vout and Vin.
What you calculate is basically 2 * sqrt(2) * ( |Vout(f)| - |Vin(f)| ), i.e. the former, scaled to peak-to-peak.