Understanding High Entropy Expressions in Circuit Analysis

[The Electrician]

I guess the part of this thread that I didn't really follow was why (somewhere in posts #50 - 60) we decided that the answer will be found by evaluating the gain at 1 Hz. Probably I missed the justification, or perhaps a computer told you that 1 Hz was the "frequency of operation".

See post #30.

 

[DaveE]

See post #30.

Thanks, I did miss that. That's an OK approach since the assumption is clearly stated that there may be some error. In fact it's probably better for a quick estimate since an exact solution is a lot of work. I do like the idea of rough estimates before the analysis. Like "how many poles and how many zeros?"; "What does it do at 0 Hz or ∞ Hz?"; or "where in general are the poles & zeros".

However, I do think that you need a step at the end to estimate how bad your estimate is; i.e. was a 1 Hz estimate reasonable, how do you know it shouldn't have been 2 Hz or 1/2 Hz, for example. Maybe I'm too picky.

Given the poorly worded problem, I'd give an "A" grade for effort in this approach. It is a reasonable estimate. If you were working for me designing control systems, I'd give you a "C-", because in the real world the estimate is a great first step, but it isn't the answer. And if you could only show me a SPICE simulation without any concept of what causes the result, or without doing an estimate like this, I might fire you.

 

[peasngravy]

It's really interesting to see just how many ways there are to solve it. Seeing how Dave and the electrician both solved it differently was very cool, but also way above my level.

About the complex maths part of it - I'm not asking for an explanation of how to do this but I thought with complex mathematics you would be left with an imaginary number as part of you answer - how does that figure into the gain, or does it cancel out further down the line?

I am happy enough with the rough estimate that I have - this is just for my own interest now if you can be bothered :)

 

[The Electrician]

Because you used impedance magnitudes throughout your calculations, your final result was a simple real number.

You are right to wonder about this. The typical result for gain of a complicated circuit like this would have a non-zero imaginary part. What this means is that the gain would have a non-zero phase angle.

The complex result I got for the overall gain is -751.706 - j 2430.6
I converted it to a magnitude of 2544.19 when I reported it.

 

[DaveE]

It's really interesting to see just how many ways there are to solve it. Seeing how Dave and the electrician both solved it differently was very cool, but also way above my level.

About the complex maths part of it - I'm not asking for an explanation of how to do this but I thought with complex mathematics you would be left with an imaginary number as part of you answer - how does that figure into the gain, or does it cancel out further down the line?

I am happy enough with the rough estimate that I have - this is just for my own interest now if you can be bothered :)

The complex result in transfer functions is nearly exactly the same as complex impedances, not really any different than what you are already doing with phasors. It encodes gain (the magnitude of the complex result, √(Re² + Im²) ) and phase shift (tan^-1(Im/Re)). We use complex numbers for these calculations because it is much easier when you have to add and multiply elements (mostly addition, actually).

 

[DaveE]

I wasn't lucky enough to study under Middlebrook (not here anymore) but I'm familiar with his work. He would say that the overall gain expression is a high entropy expression.

Actually, I don't think so. He would say it could be high entropy if you describe the exact solution. But he would also champion making the appropriate approximations to focus on the salient results. The problem with this is you can only know how to make those approximations when you understand how the individual pieces relate to each other.

However, there are cases (like well designed filters) where the poles and zeros are all proximal and it is difficult to make valid approximations. In those cases there is no alternative to high entropy. The approximations can't actually make substantive changes to the result, of course.