[sdiy] Math Geniuses Wanted!!!

[Tim Stinchcombe]

Hi Jim,
	I've worked through your algebra, and apart from the sign you'd
already spotted, it looks OK to me. (Minor typos: after eqn. (13), word
missing at end of sentence; penultimate sentence, not -> note; several
places 'IB' rather than 'I sub B'. Also for clarity, somewhere around (18),
(19): x=Vin, then (23) is dIo/dVin=d/dx(u/v)=...)

However, there is a much neater way (as you suspected), which I've taken
from 'Analog Circuit Techniques' by TH Wilmshurst, Newnes, 2001 (a kind of
MSc 'conversion course' type book, so it is very terse...). In outline:

Divide (3) into (2), to get I1/I2=e^{Vin/VT}.
Sub for I1 from this into (1) to get I2=IB/(1+e^{Vin/VT}).
Rewrite as: (IB/2)*(2/(1+e^{Vin/VT}))
=(IB/2)*(1+e^{Vin/VT}+1-e^{Vin/VT})/(1+e^{Vin/VT})  (similar to your
'fiddle')
=(IB/2)*(1-tanh(Vin/(2*VT))).
Similarly for I1, but this time you get '+tanh'.
Subtracting gives Io=IB*tanh(Vin/(2*VT)), a much nicer expression to handle.
Taylor expansion of tanhx is x-x^3/3+..., so the linear approx is just
IB/(2*VT) or 19.2*IB as you have.
Derivative d/dx(tanhu)=sech^2(u) * du/dx, giving
IB/(2*VT)*sech^2(Vin/(2*VT)).
A quick substitution for the 'e^x' expression for sechx appears to give me
your Gm expression.

Whenever I'm messing with stuff like this, I find the 'Mathematical Handbook
of Formulas and Tables' by Murray R. Spiegel, Schaum's Outline series,
McGraw Hill (care: the 'full' version, not the 'abridged' one!) contains
most expressions, expansions, relations etc. that I need.

Cheers,
	Tim
__________________________________________________________
Tim Stinchcombe 

Cheltenham, Glos, UK
email: tim102 at tstinchcombe.freeserve.co.uk

 
>     I have been doing a little OTA analysis stuff, trouble 
> is, I have a minor error in some of the math that I can't figure out.
> 
> http://www.silcom.com/~patchell/SecretsOfOTAs.pdf
> 
>     If you look at the above document, I attempted....oh 
> wait, I see my error now....DAnG...well, good 
> actually....there is an error in equation (23), (an extra 
> minus sign)...but that was because I failed to notice the 
> minus sign in equation (16)....however, as long as I posted 
> it, can anybody else find any other math errors....it has 
> been, after all, 30 years since I have done any calculus....
> 
>     I need to get a good book on calculus to use as a 
> reference...(I gave my text I had in college to a friend that 
> could not afford to buy one, back then the calculus text cost 
> $10!, most because I was so good at it I figured I would 
> never forget how to do it, WRONG!)...any recommendations?
> 
> --
>  -Jim
> ------------------------------------------------
> * Visit:http://www.silcom.com/~patchell/
> *-----------------------------------------------