Noise and gain in transimpedance amplifiers

There are some useful references available online originally from the Burr-Brown Applications Handbook (my 1994 copy). They are now also available from the TI website (TI.com). There are three that may be of use:
Noise analysis of FET transimpedance amplifiers: AB-076, or TI sboa060
Photodiode monitoring with opamps:AB-075, or TI sboa035
Compensate transimpedance amplifiers intuitively: AB-050, or TI sboa055a

Hope these may be of some assistance. Scott.

OK, I've sorted out my error in thinking.  

I had in my mind that:
  e_out = i*Rf 
But of course that's wrong.  The output voltage has a constant term as well:
  e_out = e_N + i*Rf

Indeed, there's no inconsistency between an "e_N*C_in" noise current that grows in proportion to frequency, with a noise gain (Avcl) that is flat at low frequency (to second order in frequency).  At low frequencies, i*Rf term is small perturbation compared to e_N, so the noise gain is flat, even though i (and therefore i*Rf) grows proportional to frequency.

Said another way:  As frequency increases from DC, the output voltage rises from e_N to (e_N + epsilon) where epsilon << e_N.   The epsilon term is indeed proportional to frequency (it's given by i_N*Rf where i_N is the current through C_in).  But epsilon << e_N for frequencies below fz=1/(2*pi*Rf*Cd), and so the noise gain stays approximately flat with frequency.  In this approximation, the noise gain is given by (e_out / e_N) ~ 1+f/fz.

For those interested, I've made some plots of the actual noise gain (Avcl = non-inverting closed-loop gain), along with this linear approximation: Avcl ~ 1+f/fz, to explore how well the linear approximation holds:

  http://academics.wellesley.edu/Physics/jbattat/electronics/tia.html

The bottom line is that the linear approximation holds very well for frequencies below 1/(2*pi*Rf*Cf).  This agrees with the discussion in AoE (Section 8.11.4, p. 539) [1], with one possible wording clarification (see footnote below).

So I'm comfortable now with thinking of the op-amp voltage noise as creating a current through C_in that is proportional to frequency.  I see that this current is linear in frequency even well above 1/(2*pi*Rf*Cf).  And I see that the noise gain Avcl is well-modeled by the linear approximation (e_N+i*Rf) for frequencies below 1/(2*pi*Rf*Cf).

Many thanks for all of your input, especially the many replies from the ever-patient Phil,
James


[1] AoE discussing the e_N*Cin current: "It would continue to rise forever, except for the effect of the parallel capacitance Cf, which causes the noise current to flatten off at a frequency fc=1/(2*pi*Rf*Cf)"

2 notes about this quote from AoE:  

(1) their fc is not the same my fc in the linked webpage above.  I use fc to represent the unity-gain frequency of the op-amp.  They use fc for the TIA usable bandwidth. 

(2) The wording in AoE implies that the e_N*Cin current levels off, but I find that the noise current continues to increase with frequency well above 1/(2*pi*Rf*Cf).  It's true that the noise *gain* levels off above this frequency because nearly all of the current goes through Cf, not Rf (see also Phil's first response in this thread, which I now understand more carefully than on my first reading!).  In other words, something does level off, but it's the noise gain, not the current through Cin.  Presumably the current through Cin should remain linear in frequency until the op-amp is no longer able to make the inverting input follow the non-inverting input.

On 12/14/2015 02:17 PM, jbattat@gmail.com wrote:
> OK, I've sorted out my error in thinking.
> 
> I had in my mind that: e_out = i*Rf But of course that's wrong.  The
> output voltage has a constant term as well: e_out = e_N + i*Rf
> 
> Indeed, there's no inconsistency between an "e_N*C_in" noise current
> that grows in proportion to frequency, with a noise gain (Avcl) that
> is flat at low frequency (to second order in frequency).  At low
> frequencies, i*Rf term is small perturbation compared to e_N, so the
> noise gain is flat, even though i (and therefore i*Rf) grows
> proportional to frequency.
> 
> Said another way:  As frequency increases from DC, the output voltage
> rises from e_N to (e_N + epsilon) where epsilon << e_N.   The epsilon
> term is indeed proportional to frequency (it's given by i_N*Rf where
> i_N is the current through C_in).  But epsilon << e_N for frequencies
> below fz=1/(2*pi*Rf*Cd), and so the noise gain stays approximately
> flat with frequency.  In this approximation, the noise gain is given
> by (e_out / e_N) ~ 1+f/fz.
> 
> For those interested, I've made some plots of the actual noise gain
> (Avcl = non-inverting closed-loop gain), along with this linear
> approximation: Avcl ~ 1+f/fz, to explore how well the linear
> approximation holds:
> 
> http://academics.wellesley.edu/Physics/jbattat/electronics/tia.html
> 
> The bottom line is that the linear approximation holds very well for
> frequencies below 1/(2*pi*Rf*Cf).  This agrees with the discussion in
> AoE (Section 8.11.4, p. 539) [1], with one possible wording
> clarification (see footnote below).
> 
> So I'm comfortable now with thinking of the op-amp voltage noise as
> creating a current through C_in that is proportional to frequency.  I
> see that this current is linear in frequency even well above
> 1/(2*pi*Rf*Cf).  And I see that the noise gain Avcl is well-modeled
> by the linear approximation (e_N+i*Rf) for frequencies below
> 1/(2*pi*Rf*Cf).
> 
> Many thanks for all of your input, especially the many replies from
> the ever-patient Phil, James
> 
> 
> [1] AoE discussing the e_N*Cin current: "It would continue to rise
> forever, except for the effect of the parallel capacitance Cf, which
> causes the noise current to flatten off at a frequency
> fc=1/(2*pi*Rf*Cf)"
> 
> 2 notes about this quote from AoE:
> 
> (1) their fc is not the same my fc in the linked webpage above.  I
> use fc to represent the unity-gain frequency of the op-amp.  They use
> fc for the TIA usable bandwidth.
> 
> (2) The wording in AoE implies that the e_N*Cin current levels off,
> but I find that the noise current continues to increase with
> frequency well above 1/(2*pi*Rf*Cf).  It's true that the noise *gain*
> levels off above this frequency because nearly all of the current
> goes through Cf, not Rf (see also Phil's first response in this
> thread, which I now understand more carefully than on my first
> reading!).  In other words, something does level off, but it's the
> noise gain, not the current through Cin.  Presumably the current
> through Cin should remain linear in frequency until the op-amp is no
> longer able to make the inverting input follow the non-inverting
> input.
> 

Another way of looking at it is that the rising i_N contribution gets
flattened out by the Rf*Cf rolloff.

The reason the noise gain is flat to first order (not to second order)
at low frequency is that the linearly growing C_d term is in quadrature
with the constant term.

 |1 + j omega C_d | = sqrt(1 + (omega C_D)**2) ~ 1 + 0.5*(omega C_D)**2
+ h. o. t.

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

> Another way of looking at it is that the rising i_N contribution gets
> flattened out by the Rf*Cf rolloff.

Agreed.
 
> The reason the noise gain is flat to first order (not to second order)
> at low frequency is that the linearly growing C_d term is in quadrature
> with the constant term.
> 
>  |1 + j omega C_d | = sqrt(1 + (omega C_D)**2) ~ 1 + 0.5*(omega C_D)**2
> + h. o. t.

I may have used a term incorrectly.  When I say "flat to second order" I meant that the noise gain is flat (at unity) unless you consider terms that are second order (and higher) in frequency (as in your expansion above). 

By the way, in your first post, you alluded to a 3rd edition of your text.  Is the timeline for that public/known yet?

>By the way, in your first post, you alluded to a 3rd edition of your text. �Is the timeline >for that public/known yet? 

Nope. Probably 2017 sometime--I'm down to a couple of hundred FIXMEs, but there's some condensing to be done so that it doesn't wind up being a phone book.

I've been scribbling things in there pretty continually since 1994.
  

Cheers

Phil Hobbs