ElectroYou - la comunità dei professionisti del mondo elettrico

da **DarwinNE** » 23 dic 2011, 17:17

Hello to everybody,
stimulated by the discussions with

DirtyDeeds, 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...

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!

da **DirtyDeeds** » 26 dic 2011, 0:46

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 $v_\text{b}$ .

da **DarwinNE** » 26 dic 2011, 13:31

DirtyDeeds ha scritto:Well done, Davide

Many thanks! I am happy you enjoyed

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

Oh, 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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Linear algebra and electronics

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