# Type 1 Error Amp Analysis

#### Andrew Pikul

Joined Mar 30, 2016
3
FOR THE LIFE OF ME I can't prove that the Type 1 error amplifier works the way my textbook claims it does. (figure 1, pg2)
It just looks like two voltage dividers, Vref = Rb / (Zc + R4) = R4 / (R1 + R4). But I can't factor out the R4 and I'm thinking it's because I'm trying to do AC analysis on the amplifier which is essentially an unknown circuit but I want to be able to derive this stuff myself.

#### shteii01

Joined Feb 19, 2010
4,644
FOR THE LIFE OF ME I can't prove that the Type 1 error amplifier works the way my textbook claims it does. (figure 1, pg2)
It just looks like two voltage dividers, Vref = Rb / (Zc + R4) = R4 / (R1 + R4). But I can't factor out the R4 and I'm thinking it's because I'm trying to do AC analysis on the amplifier which is essentially an unknown circuit but I want to be able to derive this stuff myself.
Like WBahn says: check your units.
You have volts on the left, ohms on the right. How do you expect ohms to magically transform themselves into volts?

#### Andrew Pikul

Joined Mar 30, 2016
3
Okay sorry, that's was typo that came from me writing this at 2am frustrated as I want to bed, it doesn't fundamentally solve the issue.
The issue is, when I derive the transfer function, I get
It's Vout/Vin = (R1+R4) / (R4+Zc)

And they say it's -Zc/R1

I have some fundamental misunderstanding of this circuit.

#### Jony130

Joined Feb 17, 2009
5,457
First, do you understand how inverting amplifier works ? And that the -V = +V = 0AC ??And this is why they omit R4.

#### Andrew Pikul

Joined Mar 30, 2016
3
Ah thank you!

I never thought about a DC ground vs AC ground. I'd like to read more about this if you have a recommendation.

I actually haven't done much AC analysis on circuits with amplifiers (but I did study high frequency response of BJT's and FET's).

#### MrAl

Joined Jun 17, 2014
10,620
Hello there,

There is a difference between an approximation and the factoring of an exact analysis. Sometimes what we do is we know beforehand that an approximation works so we apply that without proof, then derive the equations from that. Doing it that way could lead to a completely different looking solution while the numerical solutions for both the exact analysis and the approximation come out very close to the same value.

Without reading the whole text, in this case it looks like they applied a known approximation and then derived the resulting equation(s) from that approximation which involves assumptions that would have also been known beforehand. This means if you try to do a truly exact analysis and try to match their solution you may never be able to do it because you did not take their assumptions into account.

What you have to do if you dont get the same solution then is to try to figure out how they derived the approximation(s). This usually involves trying to find out how some variable becomes either completely wiped off the map (factored out) or just becomes negligible with respect to the other quantities that come out of the analysis. For this problem it is most likely the latter.

For example, if we do an exact analysis and come up with a solution with two terms summed y=A+B and A is 10 and B is only 0.001, it seems clear that an approximation would be y=A. Note we did not factor out the B, we just completely eliminated it by virtue of it's typical numerical value with respect to the other more significant values in the solution. My guess is for this problem you have to do that. That would involve doing an exact analysis and then breaking it up into separate terms and trying to figure out if the terms with R4 in them become insignificant for most applications or at least the one you are most interested in.

Of course factoring out is easy if we have something like this:
s*R1*R4/(s*R2*R4+R4)

as we just factor out the R4 and get:
s*R1/(s*R2+1)

but other terms will involve examining them carefully to figure out if they can be eliminated because they are too small to worry about. A really simple example would be:
s*R1/(s*R2+R4^9)

This is an extremely sarcastic example because R4 is brought up to the 9th power which we usually dont see much, but what this means is that if R1 and R2 are around 100k and 10k for example and R4 is only 1k, R4 taken up to the 9th power is a huge number which drops the entire term numerically down close to zero, and although that in itself is not enough, if it is added to another term such as Vref, and Vref is as little as 1 volt, we see this term affects the total solution very little so we think about ignoring it for the approximation. We check to see that R1 does not become comparable to R4^9 too or else we might not get away with it, but these are the kinds of things we look for when pure factoring is not an option.

Keep in mind this is not the same as cheating or assuming anything negligently without really knowing how it works, it is still a true analysis but we carefully determine through real applied math what we can ignore. This is not the same as saying, "Do this or do that just because you can do it"