Noise and gain in transimpedance amplifiers

It's asymptotically 3 poles because of the rolloff of A_VOL. I could have put that better, it's true.

No, that doesn't help the input current noise, it just cuts down the gain seen by the higher-frequency noise currents.

Well, because a gain analysis isn't a noise analysis, for one thing.

Not if you don't put it in. SPICE will get it right, except of course that all the op amp models you can get are horrible.

It isn't exactly Cd*e_N, because at higher frequencies the op amp doesn't have enough gain to hold the summing junction exactly still (its idea of still, i.e. impressing its input noise voltage across C_d). But it's close enough for noise purposes.

Gain analysis and noise analysis are different. They're often confused, because we're very accustomed to noise sources that are white, e.g. Johnson noise and shot noise. 1/f noise goes up at low frequency, and e_N*C_d noise goes up at high frequency. They don't have the same shape as the gain curves at all. Above the 1/f region, white noise dominates until the differentiated voltage noise becomes large.

Yup. Input-referred means "what input signal would generate this output, given a noiseless amplifier with the same gain?"

Good on you for drilling through all that. It's amazing to me how many people refuse to do their own math, even when their livelihoods are riding on the success of their projects. In optical measurements, it's easy for one's intuition to be off by 5 orders of magnitude.

And calculating this stuff is fun once you get used to it. I did a TIA for a biochip application a few years ago (for the research arm of a Very Very Large Korean Electronics Company). It got within (iirc) 6 dB of the shot noise limit for a 1-nA signal out to 70 MHz. It used a cascoded pHEMT front end and a super small wire bonded input connection so that the total input capacitance was about 0.7 pF. It used capacitive feedback for AC and an active current source for DC, and fixed up the resulting frequency response funnies in the third second stage.

I definitely couldn't have done that without doing the math first. (Of course 1 nA is only 16 dB above its own shot noise in 70 MHz, and the signal couldn't be made repetitive, so it was a pretty marginal measurement to begin with.)

Cheers

Phil Hobbs

On Thursday, December 10, 2015 at 12:05:59 AM UTC-5, snipped-for-privacy@gmail.com wrote :

er. I've been mainly reading Hobbs (2nd edition) and H&H AoE (3rd Ed). Re ferences below are to those editions. Most importantly, thank you to the au thors for these wonderful texts!

that the very steep rolloff is equivalent to 3 poles. I'm trying to under stand why the roll-off is 3-pole and not 2-pole (I may be missing something obvious here). In particular, the denominator of Zm is a quadratic in s, so shouldn't there only be two roots (poles)? Also, Hobbs' plot of Zm (Fig . 18.5) shows a 12dB/octave rolloff, not 18dB/octave. e.g. from that plot, Zm(1MHz) = 5,200 ohm and Zm(2MHz)=1,300 ohm, and 20*log(5,200/1,300) = 12dB.

t (e.g. AoE 8.11.3, and in a May 4, 2015 post to this group by Winfield Hil l [1]), and I gather that this problem is mitigated at f >> 1/(2pi Rf Cf) b ecause Cf shunts Rf.

uded "by hand". In other words, shouldn't e_N-Cin noise naturally fall out of an analysis of Avcl (the non-inverting closed loop gain)? Specifically , if Avcl is used to find the output-referred noise voltage, which is then converted to an input-referred noise current, shouldn't that noise current include the "e_N-Cin" contribution?

gives an expression for Avcl, and the following sentence says "For frequen cies well within the loop bandwidth, the resulting equivalent noise current is approximately i_n = (2pi f Cd) e_N."

equency (see Hobbs Fig 18.5), so won't the resulting input-referred noise c urrent also be flat (not rising with frequency, as required by the e_N-Cin noise)?

put noise current. Which brings us to the next and final question that wil l further reveal my ignorance ...

e_N times Avcl) to the input-referred noise current density, do you simply divide the output noise voltage by the transimpedance: i_Nin = e_o/Zm?

ge noise on the summing junction causes it to move up and down, and the cap acitance on that node needs current to do this, which is supplied by the op

-amp's output, and appears as signal."

s)

Good I see Phil responded. S. Franco in "Design with Operational Amp...." does a nice job of doing the noise analysis in the TIA photodiode circuit. (Lots of other good stuff too.)

As Phil says it's good to work through it yourself. With Franco you've got someone to hold your hand.

George H.

Thanks for the reply Phil. I'm really enjoying working through these TIA c alculations. Stretches the mind. Always good to get some help from expert s, too -- some follow-up questions below:

I did include A_VOL to get the quadratic. A_VOL gives a factor of 1/s and so the quadratic comes from the product of A_VOL and Z_f (which goes like 1 /(1+s)). I don't see where the cubic comes from.

Ah -- I think I now know. Below fc, Aol ~ 1/s, but above fc, Aol has a 2-p ole roll-off, so the Aol ~ 1/s assumption eventually breaks down. Is that what you mean by "asymptotically 3 poles"?

If so, then that makes sense to me, and is consistent with Fig. 18.5. You' d expect to see the transition from 2-pole to 3-pole rolloff above 2MHz, wh ich is outside of the frequency range for Zm shown in Fig. 18.5 (assuming a n LF356 with fc=4MHz, as indicated in Fig. 18.4).

But the "noise gain" *is* Avcl. (i.e. Avcl is dVout/dVin, measured at the noninverting input). e.g. from the discussion leading up to Hobbs Eqn 18.1

Or do you mean something different by "noise analysis?"

OK, this is getting to the heart of the question. I'm arguing that I have put it in. I compute the noise gain (Avcl) for e_N by applying Kirchoff's Current Law at the summing junction (inverting input). Current through Cin is included in this model. Even the current driven through Cin by the op- amp voltage noise. Why would I need to add, separately, another contributi on "e_N-Cin" current? Avcl *does* account for all of the current through C in that is caused by e_N.

See derivations in URL below, along with claim that the noise current is on ly proportional to frequency (i.e. the so-called "e_N-Cin noise") in a limi ted region of frequency space, between fz=1/[2*pi*(Cin+Cf)] and fp=1/(2

Yes, this derivation (1) ignores the rolloff at very high frequency caused by op-amp open-loop gain rolloff, and (2) assumes white op-amp noise voltag e. But these assumptions can easily be included -- and they don't change m y basic conclusion.

What I meant was that the book implies that 18.12 comes out of 18.11 (i.e. that indeed the Avcl expression contains the phenomenon of "e_N-Cin noise") . Which I agree with.

I'm on board with the frequency dependences of Johnson, 1/f, shot, etc. Bu t the expression for e_N-Cin noise states that this noise current grows lin early with frequency even for white op-amp voltage noise (e_N). I'm trying to find the physical origin of this "effect" and claim that it's fundament al origin is the linear rise of the noise voltage with frequency (over a li mited range of frequencies).

James

Thanks for pointing out this reference. I hadn't seen it and look forward to reading it. I've found it very helpful to read about the same idea from multiple sources (so far Hobbs, AoE and Graeme have been very complementary).

Once the op amp isn't holding its input still any longer, the diode capacitance makes Vin go as 1/f, which gives you an extra pole.

I clarified the discussion in the draft third edition, thanks!

OK, there are two gains in the problem--the noise gain and the transimpedance gain. I agree that the noise gain is A_VCL (noninverting).

Sure. I was thinking of the transimpedance as the gain under discussion.

It does. What I'm trying to get across there is that the noise current going through C_d is a real current that you could in principle measure with a meter. Multiply that by the transconductance, and you get the C_d*e_N contribution to the output noise.

It's a way of supplying physical motivation for what the A_VCL calculation predicts.

I'm trying to find the physical origin of this "effect"

Cheers

Phil Hobbs

Thanks Phil, very helpful responses. My apologies that my first question a bout the transimpedance gain (Zm) created confusion about what gain I was u sing for the noise analysis.

On the "e_N-Cin noise" question, it seems like we're in agreement then that the rise in input-referred noise current density with frequency is indeed caused by the ramp up of Avcl between fz and fp, and since that happens ove r a finite range of frequencies the "e_N-Cin noise" (meaning a current prop ortional to frequency) only exists over that limited range of frequencies. i.e. the "e_N-Cin noise" is not a separate phenomenon -- it's a consequenc e of the shape of Avcl.

Below fz, the input noise current is white (until you go down in frequency far enough to reach the 1/f rise in e_NAmp -- e.g. ~100Hz for the LF411), a nd above fp the input current noise is also white (until you get hammered b y the rolloff of Zm and/or Avol).

James

t the rise in input-

up of Avcl between

e_N-Cin noise"

d range of

t's a consequence of

It's an input vs output question. e_N causes a real current to flow in the external circuit, more or less independent of the details of the feedback n etwork. That means that its effect on the SNR (which is what we mostly care about) hardly depends on the feedback details. I'm in favour of keeping it that way and multiplying it by the transconductance, because that keeps cl oser to the physics and makes it much easier to compare different designs i n my head.

So no, I wouldn't agree that A_VCL causes the effect. I'd say that C_d caus es both A_VCL and the e_N*C_d noise, but that the latter is more fundamenta l.

Cheers

Phil Hobbs

e external circuit, more or less independent of the details of the feedback network. That means that its effect on the SNR (which is what we mostly ca re about) hardly depends on the feedback details. I'm in favour of keeping it that way and multiplying it by the transconductance, because that keeps closer to the physics and makes it much easier to compare different designs in my head.

uses both A_VCL and the e_N*C_d noise, but that the latter is more fundamen tal.

Now you've lost me. Maybe because of semantics, maybe something more funda mental. I don't see how e_N*C_d noise is a separate phenomenon from A_VCL (which already accounts for current flow through C_d).

Let's think about the low-frequency limit here. And let's simplify the fee dback network to be a resistor (the impedance of Cf at low frequency is ver y high anyway). Let's also assume that eN is white for now.

In that scenario, if the current through C_d is indeed i ~ 2pi*Cd*eN*f, the n this current must also flow through Rf, to produce an output voltage nois e density that grows linearly with frequency. But that conflicts with the result that I think you agree with: e_out = Avcl*e_NAmp which implies that e_out is white, because e_NAmp is white, and Avcl is fla t at low frequency. Do you agree that these two analyses disagree (current through C_d ~ f, vs. e_out = Avcl*e_N)?

This is one of those times when a brief conversation at a whiteboard is pri celess! Thanks for your patience -- I'm learning a lot here.

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.

Cheers

Phil Hobbs

Yes, I agree that Avcl changes when you change Cf (well, above f=1/(2pi*R f*Cf). Below that frequency, Avcl is unity independent of Cf). But the SN R 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 t o 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 freque ncy from DC on up to fp, then at low frequency the output noise voltage wou ld 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.

Cheers

Phil Hobbs

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.

Cheers

Phil Hobbs

...snip...

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.

Also, can you comment on the low-frequency hypothetical?

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.

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.

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.

If you're more comfortable with the closed-form expression for A_VCL, that's certainly OK with me. I find that keeping close to the physics makes it a whole lot easier to do tradeoffs and find new topologies. There's lots of life after op amp TIAs. ;)

Cheers

Phil Hobbs

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

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.)

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. ;)

Cheers

Phil Hobbs

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 cu rrent 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 frequ ency). But that cannot be right since e_output = Avcl * e_NAmp and the t wo 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.

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.

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?

Some folks like to tie everything up in one neat mathematical expression (radar books are full of those, for instance). I'm not in that camp. What I mostly care about is knowing what will happen if I change things, because I do that a lot. Hanging on to things that have to be true, e.g. the physical origins of the noise, is a big help.

Cheers

Phil Hobbs

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.