Noise and gain in transimpedance amplifiers

On 12/10/2015 10:07 PM, George Herold wrote:
> On Thursday, December 10, 2015 at 9:35:55 PM UTC-5, Phil Hobbs wrote:
>> On 12/10/2015 09:29 PM, jbattat@gmail.com wrote:
>>>>> SNR = i_signal * Zm / (e_N*Avcl)
>>> ...snip...
>>>> You left out the signal.
>>>
>>> Why do you say I left out the signal? I wrote down the SNR as a ratio
>>> of voltages at the output.  The signal is i_signal * Zm  (i_signal is
>>> the photodiode current).  The noise (again, voltage at the output) is
>>> e_N*Avcl.
>>
>> I clarified that in my follow-up.  Once you mix the signal and noise
>> currents together, there's no getting them apart again.  The physics is key.
> Well, you can do those correlation things.. multiplying two "equivalent"
> signal chains can get rid of the uncorrelated amp noise... slowly.
> It's a case when the noise is the signal. :^)
> (I've never done it.... seems like a lot of work.)
>
> George H.

In cases where you can make an ensemble of multiple instances of the 
same random function, sure.  The classical example is using a bunch of 
noisy MOSFETs to measure a quiet resistor, by (as you say) 
cross-correlating their outputs pairwise and averaging the result. 
MOSFETs don't have any low frequency current noise to speak of, so the 
measurements are independent.  Given enough time, you can make a good 
measurement measure maybe 20 dB below the FET noise floor.

However, the case under discussion is a single instance, where there's 
no way to distinguish signal from noise once they're mixed together. 
(Of course if I'm wrong about that, and you find a good general method, 
you can make your fortune.)

Cheers

Phil Hobbs


-- 
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

> It isn't inconsistent.  A_VCL is asymptotically flat at low frequency 
> (neglecting 1/f noise), but that doesn't mean it's exactly flat.  The 
> e_N*C_d noise doesn't magically disappear when it drops below the white 
> noise floor.

It *is* inconsistent.  the "e_N*C_d" noise predicts that e_out grows linearly with frequency.  e_N*A_VCL predicts that e_out is flat to second order in frequency.  

Also, what "white noise floor"?  There are not two separate noise sources here.  Just e_N.

> With all due respect, it seems to me that you don't care enough about 
> the physics of the problem.  What's so hard to understand about the 
> origin of the e_N*C_d contribution?  Is it too untidy?

No offense taken, but I'd actually argue the opposite.  To me, the "physics of the problem" means finding the root cause (like the shape of Avcl) that explains a bunch of individual "effects" (like e_N*C_d noise).  At it's core, there's just an input noise voltage and a voltage gain.  It may be more handy and quick to keep the term "e_N*C_d noise" in your pocket, but it is not the "physics of the problem".  e_N*C_d noise doesn't extend over an arbitrary range of frequencies.  It's a label to place on the shape of Avcl between fz and fp.

>   Hanging on to things that have to be true, 
> e.g. the physical origins of the noise, is a big help.

I couldn't agree more.   But e_N*C_d noise is simply not fundamental to the problem.

It's like saying that there are "hyperbolic" and "elliptical" effects in gravitational orbits, when really there's just *gravity*, which in a two body system can produce either hyperbolic or elliptical orbits, depending on the total energy of the particle.  In this op-amp problem, I'd say that there's just e_N and a voltage gain Avcl, and depending on your frequency, the equivalent input current is either flat or rising with frequency.

> No offense taken, but I'd actually argue the opposite.  To me, the "physics of the problem" means finding the root cause (like the shape of Avcl) that explains a bunch of individual "effects" (like e_N*C_d noise).  At it's core, there's just an input noise voltage and a voltage gain.  It may be more handy and quick to keep the term "e_N*C_d noise" in your pocket, but it is not the "physics of the problem".  e_N*C_d noise doesn't extend over an arbitrary range of frequencies.  It's a label to place on the shape of Avcl between fz and fp.

How about this -- AoE explains that the "e_N*C_d noise" rolls off above fp.  This is exactly the frequency that Avcl rolls off.  All I'm claiming is that as you go down in frequency, the e_N*C_d noise also rolls off below fz.  Does my phrasing it this way help?

On Thursday, December 10, 2015 at 10:24:58 PM UTC-5, jba...@gmail.com wrote:
> > No offense taken, but I'd actually argue the opposite.  To me, the "physics of the problem" means finding the root cause (like the shape of Avcl) that explains a bunch of individual "effects" (like e_N*C_d noise).  At it's core, there's just an input noise voltage and a voltage gain.  It may be more handy and quick to keep the term "e_N*C_d noise" in your pocket, but it is not the "physics of the problem".  e_N*C_d noise doesn't extend over an arbitrary range of frequencies.  It's a label to place on the shape of Avcl between fz and fp.
> 
> How about this -- AoE explains that the "e_N*C_d noise" rolls off above fp.  This is exactly the frequency that Avcl rolls off.  All I'm claiming is that as you go down in frequency, the e_N*C_d noise also rolls off below fz.  Does my phrasing it this way help?

Oh, and the reason that the e_N*C_d noise rolls off below fz is because Avcl is flat below fz.

On 12/10/2015 10:19 PM, jbattat@gmail.com wrote:
>> It isn't inconsistent.  A_VCL is asymptotically flat at low
>> frequency (neglecting 1/f noise), but that doesn't mean it's
>> exactly flat.  The e_N*C_d noise doesn't magically disappear when
>> it drops below the white noise floor.
>
> It *is* inconsistent.  the "e_N*C_d" noise predicts that e_out grows
> linearly with frequency.  e_N*A_VCL predicts that e_out is flat to
> second order in frequency.

No, it isn't inconsistent.  The e_N*C_d contribution is one of several. 
  It isn't dominant at low frequency, but it often is at high frequency.

>
> Also, what "white noise floor"?  There are not two separate noise
> sources here.  Just e_N.

There are a whole bunch of sources.  Johnson, shot, e_N, i_N, what you 
had for breakfast....

>
>> With all due respect, it seems to me that you don't care enough
>> about the physics of the problem.  What's so hard to understand
>> about the origin of the e_N*C_d contribution?  Is it too untidy?
>
> No offense taken, but I'd actually argue the opposite.  To me, the
> "physics of the problem" means finding the root cause (like the shape
> of Avcl) that explains a bunch of individual "effects" (like e_N*C_d
> noise).  At it's core, there's just an input noise voltage and a
> voltage gain.  It may be more handy and quick to keep the term
> "e_N*C_d noise" in your pocket, but it is not the "physics of the
> problem".  e_N*C_d noise doesn't extend over an arbitrary range of
> frequencies.  It's a label to place on the shape of Avcl between fz
> and fp.
>
>> Hanging on to things that have to be true, e.g. the physical
>> origins of the noise, is a big help.
>
> I couldn't agree more.   But e_N*C_d noise is simply not fundamental
> to the problem.

Have it your way.  Clearly I'm not going to be able to help much further.

Cheers

Phil Hobbs

-- 
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

On Thursday, December 10, 2015 at 9:49:15 PM UTC-5, Phil Hobbs wrote:
> On 12/10/2015 09:35 PM, George Herold wrote:
> > On Thursday, December 10, 2015 at 9:07:09 PM UTC-5, Phil Hobbs wrote:
> >> On 12/10/2015 09:04 PM, Phil Hobbs wrote:
> >>> On 12/10/2015 08:33 PM, jbattat@gmail.com wrote:
> >>>>> Imagine putting a big variable cap in parallel with R_F.  By cranking it
> >>>>> this way and that, you can make A_VCL do whatever you want, but the SNR
> >>>>> basically stays still, because e_N and C_d stay still.
> >>>>
> >>>> Yes, I agree that Avcl changes when you change Cf (well, above
> >>>> f=1/(2pi*Rf*Cf).  Below that frequency, Avcl is unity independent of
> >>>> Cf).  But the SNR is given by:
> >>>>     SNR = i_signal * Zm / (e_N*Avcl)
> >>>> and if you fiddle Cf, you're changing both Avcl and Zm.  It's not
> >>>> obvious to me that their ratio is unchanged.
> >>>>
> >>>> It would be helpful to hear your thoughts on the low-frequency
> >>>> hypothetical from my previous post.  i.e. if the noise current really
> >>>> grows with frequency from DC on up to fp, then at low frequency the
> >>>> output noise voltage would grow with frequency.  But e_N * Avcl is
> >>>> white at low frequency.
> >>>>
> >>>> James
> >>>>
> >>>
> >>> You left out the signal.  The signal current comes in via the
> >>> photodiode, just like the e_N*C_d current.   Once they're mixed
> >>> together, all the amp can do is change the frequency response.  The
> >>> maximum SNR is fixed.
> >>>
> >>
> >> I should add that the SNR discussion earlier in the chapter, where we
> >> start with a simple load resistor and work from there, is highly
> >> relevant.  Instruments live and die by their SNR and stability.
> >> Frequency response, you can fix afterwards.
> >
> > Re: SNR, I'm not sure this is true or not. But my gut says
> > that somewhere in the source size, (PD area), light level,
> > (detector R.), signal frequency, available opamp, space.
> > That a PD reversed biased (from a clean source) into an R,
> > with a opamp looking at the R voltage is as good as anything.
> > (OK I'm thinking simple and not adding any transistor jiu-jitsu.)
> >
> > George H.
> 
> Nope.  Op amps are a good 20 dB off the pace in very many instances, and 
> even further if you really tweak things to the eyeballs. For instance, a 
> bootstrap made from a BF862 JFET and a couple of BJTs can reduce the 
> effective capacitance of a photodiode by a factor of 10**3, and replace 
> the op amp's noise with the ~0.7 nV of the BF862, with a bias current of 
> 2 pA.  They don't make op amps anywhere near that good. 
I've only done bootstraps w/ opamps.  
pHEMTs are even 
> better at frequencies above about 2 MHz, though they take a bit more TLC 
> than BF862s.
> 
> Photons are often very expensive, which makes extra design effort on the 
> front end very worthwhile.  (It's also fun, once you've done it once or 
> twice.)
> 
> That said, of course there are plenty of easy cases, where the light is 
> bright and the bandwidth smallish, and the by-the-book approach works 
> fine.  (I give 5 rules for opamp-based TIA design in Section 18.4.3 of 
> the second edition, which a number of people have told me were very 
> helpful.)

OK I guess that's my case.  I've got enough photons at some BW
such that the shot noise is above resistor noise (>50 mV of sig.)
I mean there is always a resistor somewhere. 
(For us mere mortals. :^)

Of course when you're making a measurement you typically don't 
want to throw away BW.     
> 
> I certainly don't advocate adding bells and whistles you don't need, but 
> then people don't phone me up for the simple ones. ;)
Sure,  But many of us design in the simpler world, 
your bread and butter is my bell and whistle. 

George H.  
  
> 
> Cheers
> 
> Phil Hobbs
> 
> 
> -- 
> Dr Philip C D Hobbs
> Principal Consultant
> ElectroOptical Innovations LLC
> Optics, Electro-optics, Photonics, Analog Electronics
> 
> 160 North State Road #203
> Briarcliff Manor NY 10510
> 
> hobbs at electrooptical dot net
> http://electrooptical.net

> No, it isn't inconsistent.  The e_N*C_d contribution is one of several. 
>   It isn't dominant at low frequency, but it often is at high frequency.
> >
> > Also, what "white noise floor"?  There are not two separate noise
> > sources here.  Just e_N.
> 
> There are a whole bunch of sources.  Johnson, shot, e_N, i_N, what you 
> had for breakfast....

But that's just it:  in the model I'm talking about, there's only e_N.  I'm isolating a single noise term for study.  There's no Johnson, no shot, no scrambled eggs.  So unless e_N can give rise to two separate noise currents, then there's an inconsistency.

> Have it your way.  Clearly I'm not going to be able to help much further.

Quite the contrary, you've been very helpful!  I appreciate your taking the time on this thread to help me out.  I can see that I've worn out my welcome -- let me think more about this silently and perhaps bug other folks elsewhere.

Many thanks and all the best,
James

On Thursday, December 10, 2015 at 9:51:15 PM UTC-5, jba...@gmail.com wrote:
> > > Also, can you comment on the low-frequency hypothetical?
> > 
> > At low frequency, neglecting 1/f noise in the op amp, the e_N*C_d noise 
> > is swamped by white noise from the amplifier, feedback resistor, and 
> > (hopefully) shot noise.  It's still there, though.
> 
> But the point of this hypothetical is to strip away all noise contributions except e_NAmp, and to show that there's an inconsistency in the e_N-C_d current argument.
> 
> So let's ignore Johnson noise (Rf is noiseless) and shot noise (there's no signal in this example).
> 
> If there really is a noise current from e_N-C_d proportional to frequency, then the output voltage grows with frequency (since Zm is flat at low frequency).  But that cannot be right since e_output = Avcl * e_NAmp and the two terms on the right hand side are white, so e_output must also be white.
> 
> We can't bury this inconsistency in Johnson or shot noise.  The latter two don't exist in this example.

Huh... no you need to go through the math, there are two terms that 
give rise to the noise peak. Cin and the pole in the opamp gain roll off
 (It's not too hard to measure (the noise peak)
 if you build a TIA.)  

Put white noise into a high Q LC and you get a big noise peak on the output.  

George H.

OK, here's one specific question that would really help me understand the e_N*C_d current better.

The e_N-C_d noise current grows linearly with frequency.

At low frequencies the noise gain (Avcl) is flat to second order in frequency, which suggests no current flow (to first order in frequency) in the feedback network.  

So what sources the e_N*C_d current?  The op-amp inverting terminal?

On Friday, December 11, 2015 at 10:58:35 AM UTC-5, jba...@gmail.com wrote:
> OK, here's one specific question that would really help me understand the e_N*C_d current better.
> 
> The e_N-C_d noise current grows linearly with frequency.
> 
> At low frequencies the noise gain (Avcl) is flat to second order in frequency, which suggests no current flow (to first order in frequency) in the feedback network.  
> 
> So what sources the e_N*C_d current?  The op-amp inverting terminal?

OK try this,  Forget about the noise current and noise voltage.  
Imagine you've got a non-inverting amp.  R feedback, but the impedance 
from the inverting input to ground is a capacitor.  What's the voltage 
gain for signals at the non-inverting input?  

Then the noise voltage is just like an applied voltage.  

The noise current thing (e_N*C_d) is nice once you've done a few TIA's cause 
it's an easy way to estimate that piece of the noise.  

George H.