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Gain-boosting cascode_zeros (Read 3693 times)
kollayliu
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Gain-boosting cascode_zeros
Jun 15th, 2011, 12:44am
 
Dear all, how many zeros in this gain-boosting cascode and what are their expressions? Thank you!
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Gain_boosting_cascode.jpg
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raja.cedt
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Re: Gain-boosting cascode_zeros
Reply #1 - Jun 15th, 2011, 4:44am
 
hi,
3 pole and 2 zero's if your load pole is very high frequency. If o/p pole is dominate when compared to all poles like M3 ugb and pole @ cascode node then you would see mainly dominate pole only but if you analyze  carefully all other poles and zero will come as pole zero doublets.

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raj.
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kollayliu
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Re: Gain-boosting cascode_zeros
Reply #2 - Jun 15th, 2011, 7:04am
 
Dear raja.cedt

Thank you for your reply.

I wanna know is there any intuitive method to analyze the zeros of a circuit. For example, how can I quickly know how many zeros and what are their approximate expressions in this gain-boosting cascode.

Thank you very much
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raja.cedt
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Re: Gain-boosting cascode_zeros
Reply #3 - Jun 15th, 2011, 1:08pm
 
hi,
i told you the number by intuition only. Let me try to explain my though to you. At dc you will have total gain is around (gm*ro)^3. From the 3db frequency of the gain amplifier it starts falling so there is pole and when you reach ugb of gain boosted amplifier gain becomes flat and equal to gm*ro^2. Again when you reach cascode node pole  again gain starts falling and when after some time again it becomes flat. Finally when you reach O/p pole frequency gain start falling and it never come to flat ...

Just to explain you i took o/p pole is very high frequency when compared to other poles, even though it is unlikely case  you will get more insite.

Please refer the following pap.

A fast-settling CMOS op amp for SC circuits with 90-dB DC gain

Thanks.
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vivkr
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Re: Gain-boosting cascode_zeros
Reply #4 - Jun 16th, 2011, 1:10am
 
kollayliu wrote on Jun 15th, 2011, 7:04am:
Dear raja.cedt

Thank you for your reply.

I wanna know is there any intuitive method to analyze the zeros of a circuit. For example, how can I quickly know how many zeros and what are their approximate expressions in this gain-boosting cascode.

Thank you very much


Let me try to give you an intuitive guide for designing such an amp which I have used successfully on several occasions:

Your auxiliary amplifier is boosting the Zout of your main amplifier. This boosting occurs as long as the auxiliary amplifier bandwidth is larger than the main amplifier bandwidth, else you get a rolloff in Zout. When that happens, the settling time of your total amp is limited not by the main amp but the auxiliary amp. So, although you may see a -3 dB frequency which is high enough, the "REAL" gain or settling accuracy achievable in transient cases (settling in switched-cap circuits for instance) is limited by the pole-zero doublet arising from the interaction of the main and auxiliary amps.

If you want a better explanation, then read the original paper by Klaas Bult and G. Geelen. That would be from JSSC, Dec. 1990. I cannot think of a better paper than that.

Regards,
Vivek
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RobG
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Re: Gain-boosting cascode_zeros
Reply #5 - Jun 16th, 2011, 5:06am
 
Vivek - If I remember the paper Bult claimed that as long as the auxillary amplifier had less bandwidth it didn't affect the response of the main amplifier. Then I think I heard him backtrack on that in a personal conversation. I've always gone more bandwidth since that is how it falls out - what do you think? Some famous professor I know is still saying less bandwidth.

Rob
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raja.cedt
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Re: Gain-boosting cascode_zeros
Reply #6 - Jun 16th, 2011, 6:31am
 
hi rob,
assume you have c1,c2,c3 caps respectively at the drains of m1,m2,m3.

For the loop m2 and m3 if you consider stability ugb of the loop should be lower than 2nd pole. so i can say gm3/c3<gm2/c1.
When you consider entire this amplifier in loop now UGB of the loop should be lesser than first non dominate pole. so gm1/c2 <gm3/c3.

Unfortunately i don't have that paper rite long back i read that, but i feel boosting op amp ugb should be higher than overall op amp UGB.

Thanks.
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fonseca.ha
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Re: Gain-boosting cascode_zeros
Reply #7 - Jun 17th, 2011, 12:51am
 
About an intuitive way to get the number of poles and zeros. The number of poles is normally the same as the number of nodes. The number of zeros is normally equal to the number of direct branches between the input and output.
Regards,
Humberto
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vivkr
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Re: Gain-boosting cascode_zeros
Reply #8 - Jun 17th, 2011, 5:14am
 
RobG wrote on Jun 16th, 2011, 5:06am:
Vivek - If I remember the paper Bult claimed that as long as the auxillary amplifier had less bandwidth it didn't affect the response of the main amplifier. Then I think I heard him backtrack on that in a personal conversation. I've always gone more bandwidth since that is how it falls out - what do you think? Some famous professor I know is still saying less bandwidth.

Rob


Rob,

I would need to read that paper but I always thought that Bult mentioned the following criteria:

main amp BW < aux amp BW < main amp second pole

The first side of the inequality takes care of the doublet and the second one of the stability. Bult's notes (I had them from a course he gave at EPFL go into great detail on these fine points).

Anyway, I cannot imagine that you could get away with aux amp BW < main amp BW. Let's take a crude case for discussion.:

Adc, main (unboosted) = 60 dB = gm(1mS) x 1MOhm
Ro,main amp = 1MOhm
Timeconstant of main amp in loop = 1us
Timeconstant of aux amp = 10us (I am deliberately taking a wild case).
Gain of aux amp = 60 dB say

Now, at the start of a settling phase, the virtual grounds of the main amp would normally start to converge towards a precision level of 60 dB at the rate of 8.6 dB/1us, whereas the auxiliary amp virtual grounds start to settle at the rate of 8.6 dB/10 us.

At the end of about 7 us, the main amp inputs are precise to 60 dB, whereas the aux amp inputs are settled only to an accuracy of about 6 dB. If you recall the formula for gain boosting of the output impedance, your effective output impedance is now 1MOhm x 6 dB = 2MOhm. Your gain boosting factor is 6 dB, and not 60 dB!!

As for famous professors arguing the other way, the only thing I can say is that the Rout, boosted = Rout,unboosted x (1 + Aboost). So, if you cannot show me that Aboost has reached its maximum value, I cannot really understand why there ought to be any boosting. NOTE: If you are not considering transient response but steady state response, i.e. you are only applying a small signal, then you do see the boosting. I've had similar discussions with people from academia that sometime insist that the DC gain simulated from an AC analysis is all that matters, but all practising engineers, particularly those dealing with switched systems like ADCs etc. understand this difference between step and steady state response well. So no use having 120 dB DC gain if your settling time constants, both in the main and aux amp are not going to let you achieve the necessary accuracy in the amount of time you have available. I can rest my case on the excellent match between measurement, simulations and understanding of the different ADCs I've designed all based on this one concept, most of them using gain-boosted OTAs.

Vivek

N.B: Gain-boosting was already introduced in the late 70s (I don't recall by whom but it was a European paper in JSSC), but was limited to use in current mirrors for DC-type of applications. My guess is that most people did not appreciate the importance of the concepts laid out in Bult's paper or did not really understand the potential of gain boosting for building high accuracy circuits, despite a paper on settling behavior in the presence of doublets by Gray and his student (Kamath) in JSSC long ago (to be fair, that paper is full of equations but not very inspiring or intuitive, atleast to me).
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« Last Edit: Jun 17th, 2011, 6:49am by vivkr »  
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RobG
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Re: Gain-boosting cascode_zeros
Reply #9 - Jun 20th, 2011, 5:09pm
 
FWIW, the paper is here: http://courses.engr.illinois.edu/ece483/bult.pdf

I'm pretty sure the paper says the bandwidth of the auxiliary amp should be less (see the figures), but the feeling these days (by Bult and others) is that the bandwidth of the auxiliary should be more. (This is just an interesting aside/trivia. You never know what Alex will ask on Jeopardy!)

I can't believe that paper was published over 20 years ago!
rg
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nrk1
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Re: Gain-boosting cascode_zeros
Reply #10 - Jun 20th, 2011, 8:38pm
 
* main amp BW < aux amp BW

Slight correction to the above(maybe this is what was meant):
Closed loop BW of the amp < aux amp unity gain frequency

The aux amp is in unity feedback. The main opamp may not be. So it is possible for main opamp unity gain frequency to be more than the aux amp unity gain frequency and still things to work out fine. The closed loop bandwidth of the circuit using the main opamp is (feedback fraction) * (main opamp unity gain frequency). In Fig. 3 of the paper, the unity gain frequency of the aux amp is less than the unity gain frequency of the main amp. But Fig. 5 shows that the former is more than the _closed loop bandwidth_ of the main amp. This is also clearly stated in the paragraph starting with "Our approach here ..." in Section IV.

* aux amp BW < main amp second pole

This inequality if not required for stability. For instance, things would be just fine if the aux amp was ideal(infinite bandwidth). There will be a pole followed by a zero due to the aux amp's unity gain frequency and because of this, the phase dips down and back up. If the maximum dip coincides with the unity loop gain frequency[ = (feedback fraction) * (main opamp unity gain frequency) ], stability will be affected, but it won't matter if it occurs at a much higher frequency.

I believe Sackinger and Guggenbuhl published this before Bult.

Cheers
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vivkr
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Re: Gain-boosting cascode_zeros
Reply #11 - Jun 21st, 2011, 12:04am
 
nrk1 wrote on Jun 20th, 2011, 8:38pm:
* main amp BW < aux amp BW

Slight correction to the above(maybe this is what was meant):
Closed loop BW of the amp < aux amp unity gain frequency

The aux amp is in unity feedback. The main opamp may not be. So it is possible for main opamp unity gain frequency to be more than the aux amp unity gain frequency and still things to work out fine. The closed loop bandwidth of the circuit using the main opamp is (feedback fraction) * (main opamp unity gain frequency). In Fig. 3 of the paper, the unity gain frequency of the aux amp is less than the unity gain frequency of the main amp. But Fig. 5 shows that the former is more than the _closed loop bandwidth_ of the main amp. This is also clearly stated in the paragraph starting with "Our approach here ..." in Section IV.

* aux amp BW < main amp second pole

This inequality if not required for stability. For instance, things would be just fine if the aux amp was ideal(infinite bandwidth). There will be a pole followed by a zero due to the aux amp's unity gain frequency and because of this, the phase dips down and back up. If the maximum dip coincides with the unity loop gain frequency[ = (feedback fraction) * (main opamp unity gain frequency) ], stability will be affected, but it won't matter if it occurs at a much higher frequency.

I believe Sackinger and Guggenbuhl published this before Bult.

Cheers


Indeed, those are important points you make, and all correct.

And yes! Sackinger and Guggenbuhl published the concept a little earlier, but I have this feeling of having seen the idea in an older paper.

Nevertheless, the application of gain boosting to building high-precision amps was pioneered by Bult & Geelen, and they also provided a detailed guideline for designing. So they may not have invented the idea first, but they understood well what it meant and where to use it.

On a sidenote, the same holds true for a paper by the same authors in JSSC Dec. 1992, where they explain their "linear MOS current division technique", something which was published long ago by Eric Vittoz and his student but Bult and Geelen refined the work and found a really good application for it. Both their papers (1990 and 1992) won the best paper award at the ISSCC if I recall, and deservedly so. Very well written!

Vivek

Vivek
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RobG
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Re: Gain-boosting cascode_zeros
Reply #12 - Jun 21st, 2011, 7:25am
 
vivkr wrote on Jun 21st, 2011, 12:04am:
nrk1 wrote on Jun 20th, 2011, 8:38pm:
* main amp BW < aux amp BW

Slight correction to the above(maybe this is what was meant):
Closed loop BW of the amp < aux amp unity gain frequency

The aux amp is in unity feedback. The main opamp may not be. So it is possible for main opamp unity gain frequency to be more than the aux amp unity gain frequency and still things to work out fine. The closed loop bandwidth of the circuit using the main opamp is (feedback fraction) * (main opamp unity gain frequency). In Fig. 3 of the paper, the unity gain frequency of the aux amp is less than the unity gain frequency of the main amp. But Fig. 5 shows that the former is more than the _closed loop bandwidth_ of the main amp. This is also clearly stated in the paragraph starting with "Our approach here ..." in Section IV.

* aux amp BW < main amp second pole

This inequality if not required for stability. For instance, things would be just fine if the aux amp was ideal(infinite bandwidth). There will be a pole followed by a zero due to the aux amp's unity gain frequency and because of this, the phase dips down and back up. If the maximum dip coincides with the unity loop gain frequency[ = (feedback fraction) * (main opamp unity gain frequency) ], stability will be affected, but it won't matter if it occurs at a much higher frequency.

I believe Sackinger and Guggenbuhl published this before Bult.

Cheers


Indeed, those are important points you make, and all correct.

And yes! Sackinger and Guggenbuhl published the concept a little earlier, but I have this feeling of having seen the idea in an older paper.

Nevertheless, the application of gain boosting to building high-precision amps was pioneered by Bult & Geelen, and they also provided a detailed guideline for designing. So they may not have invented the idea first, but they understood well what it meant and where to use it.

On a sidenote, the same holds true for a paper by the same authors in JSSC Dec. 1992, where they explain their "linear MOS current division technique", something which was published long ago by Eric Vittoz and his student but Bult and Geelen refined the work and found a really good application for it. Both their papers (1990 and 1992) won the best paper award at the ISSCC if I recall, and deservedly so. Very well written!

Vivek

Vivek


Hi Vivek Vivek,
I bet gain boosting was known since the stone age, but the cave men didn't know how to draw transistors yet Wink Certainly using an opamp in current mirrors would have been known. Widler knew everything anyway. The problem with bipolar was that beta limited the gain of a cascode (beta is a function of Vce) and putting gain around the wouldn't help. Plus gain is pretty easy to get w/ bipolar circuits.

rg
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vivkr
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Re: Gain-boosting cascode_zeros
Reply #13 - Jun 27th, 2011, 12:09am
 
true! Probably Widlar knew it about it anyway Wink It would have been pointless to try and aim for high gains in those days though as the circuits that really do need high gain are discrete-time ones which could only be realized with MOS.

Vivek
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sean.geng
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Re: Gain-boosting cascode_zeros
Reply #14 - Jul 21st, 2011, 2:10am
 
RobG wrote on Jun 16th, 2011, 5:06am:
Vivek - If I remember the paper Bult claimed that as long as the auxillary amplifier had less bandwidth it didn't affect the response of the main amplifier. Then I think I heard him backtrack on that in a personal conversation. I've always gone more bandwidth since that is how it falls out - what do you think? Some famous professor I know is still saying less bandwidth.

Rob

hi ,RobG, can u just tell the exact name of this paper?

Regards
sean
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