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Linear algebra and electronics

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[1] Linear algebra and electronics

Messaggioda Foto UtenteDarwinNE » 23 dic 2011, 16:17

Hello to everybody,
stimulated by the discussions with Foto UtentePiercarlo and Foto UtenteDirtyDeeds, I tried to write a small article about the application of the linear algebra in the analysis of the differential pair.
The idea is that the differential and common modes comes from a matrix diagonalization.
The contents of the article are not particularly new nor exciting. They might however give another point of view and I have to say that I have not yet seen a textbook presenting this mathematical point of view on the differential pair.

Here is the article:

http://www.electroyou.it/darwinne/wiki/ ... e-symmetry

I am not very good at doing calculations, so if someone can check the results... :D
Also, if you spot some typos or some grammar errors/horrors, I would be extremely happy to correct them, so we might discuss here. I am in general eager to learn from my errors, about the language, the algebra or the electronics. Can someone check the bibliography? For example, I am on vacations right now and I do not have the Sedra/Smith book here.

Finally, if you judge interesting the article I wrote in English, I would like to prepare a translation a day or another. Of course, if someone would like to translate it in Italian, I would be very happy for that and we can collaborate!
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Foto UtenteDarwinNE
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[2] Re: Linear algebra and electronics

Messaggioda Foto UtenteDirtyDeeds » 25 dic 2011, 23:46

Well done, Davide ;-)

A small step further might be the following. Given

W = \begin{pmatrix} 1/2 & 1 \\ -1/2 & 1\end{pmatrix}

one can determine a matrix A^\prime,

A^\prime = W^{-1}AW

even in the more generale case of the non-symmetrical differential pair. Indeed, in this case, A^\prime will not be diagonal. However, if the asymmetry is small, A^\prime should be approximable as (unless I'm very much mistaken...)

A^\prime \approx D+ \begin{pmatrix} 0 & e_{12} \\ e_{21} & 0\end{pmatrix}

where e_{12} and e_{21} would contain only first-order terms in \Delta g_\text{m}/g_\text{m}, \Delta R_\text{c}/R_\text{c} etc.


PS1: Please please, do not use anymore v_\text{s} for an output voltage in an English text :mrgreen:
PS2: I think you're using an uncommon notation for voltages. Taking as an example the base voltage of a BJT, a more common notation would probably be the following:

v_\text{B}(t) = V_\text{B}+v_\text{b}(t)

where v_\text{B}(t) (lower-case letter with upper-case subscript) is the total base voltage, V_\text{B} (upper-case letter with upper-case subscript) is the DC component of v_\text{B}(t) and v_\text{b}(t) (lower-case letter with lower-case subscript) is the signal. The symbol V_\text{b} can then be used for the Fourier transform of v_\text{b}.
It's a sin to write sin instead of \sin (Anonimo).
...'cos you know that cos ain't \cos, right?
You won't get a sexy tan if you write tan in lieu of \tan.
Take a log for a fireplace, but don't take log for \logarithm.
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[3] Re: Linear algebra and electronics

Messaggioda Foto UtenteDarwinNE » 26 dic 2011, 12:31

DirtyDeeds ha scritto:Well done, Davide ;-)


Many thanks! I am happy you enjoyed :D

DirtyDeeds ha scritto:A small step further might be the following.
...


Thanks for the suggestion. In reality, I already wanted to investigate what it happens when the symmetry is not perfectly respected. I would also like to see (a day or another) what it happens to the eigenvalues, to see if they conserve some sort of interpretation. Your idea is also very interesting.

DirtyDeeds ha scritto:PS1: Please please, do not use anymore v_\text{s} for an output voltage in an English text :mrgreen:


Oh, LOL! :lol:
Of course, v_\text{s} comes straight from French...

DirtyDeeds ha scritto:PS2: I think you're using an uncommon notation for voltages.
...


This is an interesting idea. I have to say that I am not always perfectly coherent with the notations I use, so I must improve this point.
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