op-amp transfer function

Several things to note:

1) These aren't equations? What is equal to what?

2) The units don't work. I suspect these are typos but, since it is repeated, it indicates a tendency to be sloppy in this regard. You have:

Vg-V1/8k - V1-Vo/(8x10^9/s) - V1/(2x10^9/s)

which, being explicit based on what you wrote, is:

Vg - (V1/8k) - V1 - (Vo/(8x10^9/s)) - (V1/(2x10^9/s))

The first and third terms are voltage, the second term is current, and the last two terms are current.

3) You have messed up the component values. Note that



I would recommend keeping specific values out of equations until the very end and use symbolic values, such as C1 or R1, instead.

Track your units! If you are going to plug in the values, then those values have units. The capacitance isn't 2x10^-9, it is 2x10^-9F. Or just leave it as 2nF. When combining units, a F is simply As/V (amp-second per volt), since an amp-second is a coulomb. The units of s are 1/sec. So your last term, for instance, would be:



which is then easily seen to be:



The units are easy to check. You are doing KCL so you want current and you have:


(check!)

 

Our posts crossed. When I posted mine, there was only t_n_k's first response in the thread.

One thing you said in the latest post bothers me:

It sounds like you've gone into guessing mode. If you think you screwed something up, go back and redo it so that you can say, "I screwed up the inversion and it should be:" Of course, since you are just learning this stuff, it is neither surprising or unreasonable that you are unsure of your steps, so don't read too much into my comment. But, by the same token, don't just flip things upside down unless you determine that flipping them upside down is what is required to correct the error you have made, not just because it makes a symptom go away.

One you have a potential solution, always do as many checks on it as you can. Always make sure the units work our correctly (which you can't do because you haven't included your units throughout your work and have therefore robbed yourself of perhaps the most effective tool in your toolbox for detecting and identifying errors) and also check if the limiting cases work out properly, such as at DC (s=0) and arbitrarily high frequencies (s=infinity).

In this case, the response at DC should either zero since there is no connection between the input and output. At infinity, it becomes an inverting unity gain buffer.

By keeping it symbolic as long as possible, you also gain other useful checks. For instance, what if the 2nF capacitor became a short circuit (C1=ooF) and the 8nF cap became an open (C2=0F) circuit (Note that I may have assigned C1 and C2 differently than you)? You would have an inverting amplifier with a gain of R2/R1 (where I have assigned R1 to the 8kohm and R2 to the 62.5kohm).

 

Up to your next to last line, things look good. But in going from that to your final line, it looks like you are trying to say that:



Were but it that simple!

So go back and try doing the algebra correctly and you should get the correct answer.

Remember those sanity checks I told you about? In particular, if C1 is removed and replaced with an open (C1=0) and C2 is replaced by a short (C2=oo), then the transfer function is trivial and can be written down by inspection as:



In your next to last line, making those substitutions yields:



which agrees with this. But making the same substitutions in your final equation results in three of the four terms blowing up toward infinity. So, if nothing else, you know that the two equations to not agree with each other.

Another sanity check was that the gain should go to zero as the frequency goes to DC. Your next to last line passes this test and your final line doesn't.

I can't stress this enough. Everyone makes stupid algebra mistakes. I made several while working this problem. Most of them made the units fail to work out properly and the others failed at least one of my sanity checks.

 

Perhaps this will make it clearer.

Look at the next to last line of the solution you posted. It is of the form:



where is everything you have in square brackets.

You are absolutely correct in that



It's how you are taking the recipricol of that is the problem.

Here is what you are doing:



This just simply isn't true. Consider a simple example:

p = 1/2 + 1/4 (= 3/4)

Using your approach, you would say that

1/p = 2 + 4 = 6

when clearly it must be equal to 4/3.

In order to use your approach, must be a single numerator over a single denominator.

 

I've just thrown my notes away, but yes, I am pretty sure that is correct.

It's pretty clear to me that you have a significant weakness in your foundational algebra skills -- that's an observation, not a put down. That weakness will only cause you more difficulty as you proceed, both with your education and in your career. Letting Maple do your thinking for you is not going to help. In fact, I would not be surprised if a big part of the reason your fundamental skills are so weak wasn't precisely because you've had access to (and been encouraged to use) things like algebraic calculators and such all your life. It is a widespread problem.

I recommend only using them for the purposes of checking the answers you produce manually whenever possible and, when you use it to get an answer that you didn't get manually, going back and working it again until you do get the same answer. Do this pretty much exclusively until you are truly comfortable with the mechanics and, even then, do the math manually on a regular basis.

Personal story: I grew up at a time when scientific calculators were just barely becomg affordable for the typical high school student. As such, my entire education up to that point had all been done by hand and I was very good at it. In college I became reliant on a calculator very quickly and one day, in the Fall of my sophomore year, realized that I had just pulled out my $300 HP41CV in order to add two simple two-digit numbers that didn't even involve a carry. It was at that point, nearly 30 years ago, that I made a vow to myself to do as much of the math manually on paper or in my head as I could as often as I could and that has served me very well to this day.

 

Oh, and another thing. I would strongly recommend you look at the sticky on making mathematical expressions using LaTeX so that people don't have to download and open files everytime you want to post just an equation or two. It's reasonable to attach a file if you have a lot of math, especially if there is some artwork that goes along with it.