Montecarlo: correlation parameter value for matched pairs?

Hello again. What is the criteria for choosing the value of the
correlation parameter value for matched transistors, when performing
Monte Carlo simulations in ADE?

For example, in the widely-known article "Recommended Spectre Monte
Carlo Modeling Methodology" (www.designers-guide.org/Modeling/
montecarlo.pdf), they use cc=0.75 for correlating two matched
resistors. What about the values for, lets say, matched transistors in
a differential pair or in a mirror? Does the correlation parameter
should be chosen depending if the transistors are multi-fingered and
interleaved? What about the cases of more than 2 transistors (for
instance, a current mirror with several copies)?

I suspect it might have something to do with the particular process
taken into account. I am using the AMS C35B4 cmos technology, but
unfortunately I haven't been able to find any suggestions for the
values of the correlation parameter that should be used...

Thanks a lot for any ideas!

Cheers,

Jorge.

 

Hi Jorge,

This is rather a tricky question :-(
It think this is more related to the experience with the process
itself. It's really something you should discuss with the process
engineers from your foundry. It might be helpful to ask for the test
structures that have been used to extract the MC matching data and
then compare them to your structure/Application ... etc. I don't think
there is a proven solid science behind this. It's rather something
related to the feeling and the appreciation of the IC designers based
on their experience on both the process and the application. I'm not
able to propose any number for this I'm afraid. The default (i.e no
correlation) is of course the most pessimistic scenario ...

That was my attempt anyway !
let's see what other chaps think about it ...

Cheers,
Riad.

 

Thanks for your reply, Riad! ...I suspected it was a tricky
question

About contacting the foundry for this information, unfortunately I
work in a university, and we have no support from them about this kind
of issues.

Still, It would be really useful to know what others designers in the
forum use in their practical work; I really doubt that everybody is
oversizing their designs by considering no correlation at all between
groups of matched transistors!

Thanks again & regards,

Jorge.

 

The more I ask about the infamous correlation parameter, the more
astonished I get... Today I asked an associate professor, which I know
has some experience in analog design (i.e. has done some working
silicon), and he was totally surprised to learn about it... he had no
idea of its existence!!!

I have also searched in the documentation of two other design kits
besides AMS (namely, TSMC & UMC), and found no references whatsoever
about how to take into account matched groups of transistors. I think
these are rather well-known and widely-used processes, why does it
seem nobody cared about documenting the proper way of performing
montecarlo simulations, taking into account matched pairs?

....I guess virtually everybody working in analog MUST do montecarlo
simulations before going to fabrication, and MUST take into account
matching on those simulations... How do they (properly) do it? :O

Puzzled (and also desperate, but that's normal ),

Jorge.

 

Jorge,

The problem is not that complicated. Say you have two devices with a parameter which is varying
randomly with a variance of s. Then the difference between the two devices will be of
v(x-y)=2*v-2*cov(x,y)=2*v*(1-cc)
where cc is correlation coefficient. so if there's no correlation, the variance of the difference
(mismatch) is sqrt(2) time larger than the variance of the individuals (m=sqrt(2)*v). With a
non-zero correlation coeff., the variance of the difference is sqrt(2)*sqrt(1-cc).

Therefore, by choosing an appropriate correlation coefficient, you can obtain the desired mismatch
distribution. Say you want a resulting mismatch variance of w, you get cc = 1 - 0.5*w/v = 1 -
sqrt(0.5)*v/m

As an example, say you have a differential pair and want to correlate the two devices. First run a
simulation with two uncorrelated devices and from the results extract the variance of the
parameter(s) you're interested into (drain current, threshold voltage, transconductance, whatever).
Then, assuming you know what variance you *expect* for these parameters, calculate the correlation
coeffs accordingly.

The real problem here is to accurately characterize the mismatch for a given process, that is, to
know what to expect. If you know that, then your problem's solved. Unfortunately accurate and
thorough mismatch data is not often available - it takes a lot of effort and time to characterize a
process for mismatch - and you have no choice but to try and estimate it. Mostly for mature
processes specifically dedicated to analog/rf applications you will find good matching data, I guess.

Considering all these problems and the efforts it takes to predict correlation coeffs for all the
matched devices in a circuit, and considering that this effort might quite well be useless unless
you have a high confidence in your process and its data, it might be an option to oversize a bit
your design. It will give you confidence and save you time and headache...

Just my two cents
Cheers,
Stéphane

 

Thanks so much for your reply, Stéphane: so helpful, as usual
BTW... where did you learned those formulas? In some course at EPFL?
I've searched the whole web for any references on the topic and found
nothing so explanatory whatsoever! Thanks again!

Regards,

Jorge.

 

Jorge, thanks.
It's just basic statistics/probability... Any book will do. Even google will spit out that kind of
information if you look for "correlation coefficient", "correlation" and that kind of keywords. Just
need to know what to look for

Stéphane.

 

Hi Jorge,

Your professor's answer does not surprise me. The number of engineers
I knew who ignore this capability is rather high.
I do agree with Stéphane's mathematics formulas. The silicon physical
reality could be slightly different though.
The Mismatch data as provided in your model cards are extracted for
well defined structures within well defined limits. Your foundry does
not guaranty any mathematical continuity if your design structure is
different from those used for mismatch data extraction.

BTW, why don't you run few MC simulation with/without correlation to
see what's the impact on your output ? We tend sometimes to argue on
very little tiny things that end up with no impact at all. You can use
a dcmatch analysis if this is possible for you, this will give you a %
of impact. MC simulations are rather time-consuming for big designs.

Please bear in mind that what we do in industry is slightly different
from what we learn in the academia. When we design a chip for mass
production, it is common to make some over-design to ensure that all
the cases are covered, we don't take any chance as we do for R&D
projects. In other words, what is meant to be couple of transistors
for a differential pair in the theory could end up with many
transistors around for any sort of trimming.

I have written few notes about some useful mathematics I did use when
designing LDO voltage regulators. You will find some stats in French :-
(
http://riad-kaced-usenet-group.goog...rRp7kbrQuQTkuyp0rhe4k7aUzfKN-m9S9niuHrq-IEXAE

But if you don't want to waste plenty of time understanding French for
4 pages of mathematics, then go for any descent 'Mathematics for
Engineers' book as Stéphane mentioned earlier. I've got one at my
office, I can send you the title/author later on if you want.

Cheers,
Riad.

 

Hi Riad, thanks a lot for your reply! I indeed checked a little and
ran some simulations in order to see the effect of the correlation
parameter in my design. The circuit in which I am working is a dynamic
differential comparator, and the problem is that...

....If I use relatively-small transistors I get too high offset
voltages due to excesive mismatch, and
....If I use large transistors, the offset is reduced but the
comparator turns too slow to communte in the alloted time.

However, if I use the infamous correlation factor, with, say, cc=0.75,
then the monte carlo results seem to be a good compromise between
offset and speed. The problem is... should I trust these
simulations? :O

Anyhow, I think I've gotten now a more clear idea about the rationale
behind this parameter and its use. Thanks a lot also for the
statistics review: my french is real-beginner level, but I am always
looking forward to improving it. And after all, once one learns
spanish (mother tongue) and italian (laboral tongue), understanding
any other romance language is very easy, I guess!

Muchos saludos,

Jorge.