theoretical question

Probably with equal gain.

Do you not wish to work out the math?

Under what weighting constraint? There is no unambiguous crossover point in that condition. Would you want output referred noise over GBW minima? (Also, in noise amplitude in the given BW, or by noise density alone?)

Tim

-- Seven Transistor Labs Electrical Engineering Consultation Website:

I'm willing to work it out, but I want to make sure I've thought the problem through thoroughly, so that the answer isn't completely meaningless!

To keep it manageable lets say maximize the SNR at the output of the cascase over the total bandwidth.

Hmm well I've done this a few times now, need a gain of 100?... cascade two x10's.

Yeah the first stage is special, there's the first stage, sometimes called a preamp, (weird name?) and then everything else.

I guess most of the time the preamp (1st stage) has other constraints, (FET input, impedance of source...) that you tend to keep the gain lower such that the over all gain/BW is not reduced.

George H.

I did the math and I think Tim W. is right. You want to the reciprocal of the gain of the first stage, A1^(-1) to be minimum to minimize noise, but bandwidth of the cascade decreases as the reciprocal of the hypotenuse of a right triangle of the gains of the two stages, I.e. 1/(sqrt(A1^2 + A2^2)). The only way to simultaneously minimize the first quantity and maximize the second is set the gains equal...

Sure, if it's some upstream gain stages, (or is that downstream?) and everything is driven by a low impedance voltage source.

I'm just saying that if it's the 1st stage in the signal chain, then it's different. (Hmm, the final output stage can be different too, it's gotta drive something different... maybe a long cable.)

George H.

Some SPICE simulation would provide a stronger handle on the possible outcome. Also, performance characteristics of different amplifiers(class A, class AB etc.,) vary, so analyzing the problem with a candidate amplifier would allow for more concrete results.

If you want to optimize two parameters, and they aren't trivially related, you need two degrees of freedom. So, no. This is a matter of having N equations, but M unknowns.

In this example, you can minimize the noise by not having any gain; you can maximize the gain and amplify the noise; you can use tricks like positive feedback to get MORE gain than a simple cascade (but with other side-effects). So, I'm pretty sure the statement of the problem is too ambiguous. You can attempt to minimize ONE parameter, and IF THERE IS STILL A DEGREE OF FREEDOM, sometimes a second. Even THAT kind of process, still depends on which is to be optimized first.

I have two degrees of freedom - the gain of the first stage and the gain of the second stage, with the assumption that the total gain must be much greater than one. I can put all my gain in the first stage to minimize noise, but I have killed my bandwidth. I can put all my gain in the second stage, and I have both maximized the noise and killed my bandwidth, so that's no solution either. Obviously there must exist some happy medium between the two extremes.

Essentially I think I'm defining a "figure of merit" as bandwidth + SNR and seeking to maximize that, rather than trying to globally minimize or maximize one or the other individually.

This sounds like is should be attacked with a spreadsheet. The inter- relationships between the various factors is already defined. Just set up some optimization problems for your spreadsheet.

?-)

No, Whit is right. You need some total gain value. You have one gain variable, then the other is set according to the total gain value so you only have one variable. You can calculate BW and noise as a function of that one setting, but you can't relate BW and noise any other way.

In order to find the gain setting that gives you the best trade off between noise and bandwidth you just define your BW/noise utility in some equation. That gives you enough equations to be able to solve them.

Sounds like a manager. Why can't you prioritise both of the tasks I set you, at once.

Simple answer is, no. The statement makes no sense. Second, the concept is irrelevant in a real, practical system.

If one noise source is 3 times less than another, it will only contribute 5% of the total noise, therefore you can ignore it. Realistically, you will use a gain greater than 3, hence the second stage won't matter. That's the point of the Friis formula.

In fact, in real life, one of the most useful gain arrangements is to have a chain of increasing size, same current gain, amplifiers (digital invertors). This is when building up a drive capability from say, 10 ff to 10 pf. A factor of X4 is typically about optimum, despite some theoretical arguments coming up with "e".

Real analog design is about approximations and compromises, not perfection.

Third point, is that a first stage might well be designed to use more current to get get a higher BW/Noise factor. The limit is:

F(power, speed, accuracy) = 0

Hands are not tied simply by BW and Noise. There is a third variable to twiddle.

Kevin Aylward B.Sc.

"Kevin Aylward" Wrote in message:

Thanks for the reply. That's why I considered it a "theoretical question, " rather than a practical one. It was just a thought experiment to help facilitate my understanding. IRL, one is of course unlikely to use a first stage LNA with the same GBW as the second stage.

But in the situation where both amps have the same GBW, it seems that if you raise the first stage gain high enough that the second stage doesn't matter, the BW will be compromised. Say I want an overall gain of 100 - for the second stage noise to be ignored as you say, the first stage gain must be (x^2)/3 = 100, x ~17.3. But the bandwidth of such an arrangement (in this restricted thought experiment) will be significantly less than if I arranged the gains as say, 10 and 10.

I understand that it is impossible to globally maximize both the noise and bandwidth in this situation. As I mentioned in another post, one would instead be seeking to maximize a "figure of merit" which is defined as the sum of the SNR and bandwidth together.

I remember some derivation involving amp cascades and "e", but I can't now remember the details. Factor of 4 in what?

Certainly so.

"Kevin Aylward" Wrote in message:

Err... you misunderstand. The first stage gain only needs to be 3 to have only a 5% effect on total noise.

The normalised noise of the 2nd stage referred to to the input of the first would be 1/3.

Total noise would be:

Vnt = sqrt(1+(1/3)^2) = 1.054

So, a 5.4% increase due to the 2nd stage, which can be ignored.

The likely hood of a 3rd stage contributing is, essentially, non existent for any competent design, despite all the academic books blowing their own trumpet by producing as big and complicated expressions as they can to impress the reader.

Sure, for such a gain arrangement.

What one would do is use more stages or more current, or both. The point is to *design* a system to meet the specs, not tie ones hand behind ones back.

In a chain of cmos inverters, each inverter is made 4 times the size of the preceding one. For example, if the first one was W=1u, L=0.18u, i.e "1u/0.18u" the second stage would be "4x1u/0.18u", the next "16x1u/0.18u". There are issues for low noise here as well. Sometimes a source follower buffer is required so that the 1st stages can be large, but not load the source.

If the ratio is larger the system is slower, or if say, only 2 times per stage, its also slower.

Kevin Aylward B.Sc.