A correction, motivated by one of Fred's rare
technical efforts apparent in this thread, is inserted
below. The OP should take note of this. I have
added a few other details and notes as well.
There is also a little humor for those who do not
take themselves way too seriously. (See "1.")
The need for this correction underscores the
importance of peer review and the value of
stating assumptions and showing enough of
the analysis to permit it to be critiqued.
[Frequency response revised and moved.]
It has come to my attention that the 3 stages are not
"like" as I had stated. The last stage has about 7.16
fewer dB of gain than derived in my pre-corrected
analysis. Also, due to the improved loop gain in the
last stage, (as compared to that earlier analysis), the
-3dB bandwidth is about 59 KHz (rather than the
49 KHz mentioned before), and well centered on the
OP's 40 KHz signal. These corrections affect the
amount of noise that should be expected at the final
stage output, as detailed below.
The bandwidths will change with a transducer in
place, due to the slightly higher loop gain that will
produce in the first stage, so it would be premature
to worry about noise bandwidth just now.
[The input noise still appears to be correct.]
If you were to short the input, then, due to the
thermal noise of the input resistor, you would
have an equivalent input noise density of about
sqrt(4 k T R) = sqrt(4 * 1.38e-23 * 300 * 1e4)
or 12.9 nV/sqrt(Hz). Adding to that the input
voltage noise of your op-amp, 16 nV/sqrt(Hz)
typically, (and adding RSS-wise), you should
expect input noise of about 20.5 nV/sqrt(Hz).
If we were going for accuracy, a smidgen could
be added from the feedback resistor, but its
contribution would be lost in the errors already
in this calculation. (Fred might be willing to
enumerate or even quantify them for you.)
[Cut incorrect gain figures and calculation.]
With the OP's stated 200K feedback R (not the
"like" 500K I mistakenly assumed), the gain is
closer to 90 dB. This results from the ideal gain
of about 94 dB (= dB(50 * 50 * 20)) with losses
of .63 dB per stage for the HPF's, .97 dB for the
first 2 stages due to their (low) loop gains of 2,
and .17 dB for the last stage with its loop gain of 5.
The output noise density is then
20.5 nV/sqrt(Hz) * 10^(90.0/20) V/V
= 648 uV/sqrt(KHz)
Multiplying by the square root of bandwidth yields
an expectable RMS output noise of 157 mV.
[Noise bandwidth adjustment unmentioned.]
With your transducer as the input, the noise
could go up or down depending on its source
impedance. That is why you would have to
measure it or take it off the datasheet to see
whether the noise you see with it connected
is what it should be.
Once the input is better characterized, it
will probably be easy to revise the first
stage a little bit to get lower noise.
I would bet good money on long odds that
the first stage can be revised to bring the
SNR up by several dB, once the input is
better known.
[corrected in place:]
As for why you do not see the noise on the
earlier stages, that would depend on your
instrument. With a typical o'scope, not being
able to see the few mV to be expected at the
2nd stage output is to be expected. And of
course, it's even harder at the 1st stage.
The comparisons suggested between the noise
to be expected and the noise you see should
be used only for limited purposes, such as
deciding to proceed further along the noise
investigation rather than looking for another
source for the noise. [1] The way to finally
evaluate noise should involve the measured
gain of the amplifier, not its nominal gain.
[1. For example, suppose you had left an
op-amp unconnected whose inputs just so
happened to float together, and stay within
about 15 uV of each other, (offset by Vos),
such that the output was miraculously active
for an observable length of time during the
few 100's of mS before they both float to the
rail, and if that op-amp was paired with the
first stage, then, if you looked at the noise
appearing on the final stage quickly enough,
(within say, 200 mS and allowing some time
for noise observation averaging), you might
see several millionths of a dB of extra noise
that: arises at the open input at the about the
same level as the other stages have (which
estimation includes input current noise); then
appears at the output of that unconnected
amplifier after a gain of about 40 dB; couples
into the input thru the typical 90 dB of channel
isolation; and competes (at -45 dB relative to
the *real* noise sources) for a position as the
top dog noise source. This would seem to be
far-fetched in the cold light of day, but late at
night, when demons are at play, who knows
what can *might* [2] happen? ]
[2. Here, "might" must be distinguished from the
false certainty that plagues us all from time to
time, some more and more often than others. ]
If you peruse the rest of this thread, (with
filters in place, I advise), you will find some
comments about your op-amp choice that
you may find useful. And if you care about
input noise and power consumption, take
suggestions about using CMOS gates as
amplifiers with a degree of skepticism.
I have no idea what that is. Since, (apparently),
you are reaching into the noise floor for signal,
it may be worth your while to post the detector
and solicit ideas for improving it.
It is also worth noting that the above analysis
predicts the RMS value of a random signal with
a Gaussian amplitude distribution. It is common
to figure on seeing 5 times more peak value, but
if you want to calculate the actual rate at which
your detector will false detect due to that noise,
you need a more sophisticated approach based
on the Gaussian statistics.
A tighter
bandpass filter could do some good as well.
A good place for that filter would be after the
first stage. If you use an active filter, that will
be a necessity. If you use an LC filter, that will
be prudent due to coupling considerations and
noise performance issues.
P.S. to Fred: If you reply, please try to take a
rational approach to this, or, failing that, come
up with some new names and more imaginative
invective. Your latest efforts have become so
repetitious that they are really quite boring.
