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Date: Wed, 12 Jun 2024 00:05:52 -0400
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Subject: Re: Transfer function reduction math
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On 6/11/2024 11:17 PM, Don wrote:
> bitrex wrote:
>> Don wrote:
>>> bitrex wrote:
>>>
>>> <snip>
>>>
>>>> Seems like they're doing some kind of tanh interpolation but it's not
>>>> entirely obvious to me how they get from equation (3) to the expression
>>>> in (5).
>>>
>>> Unless the fuzzy form of your scan deceives my eyes, it appears the
>>> numerator and denominator are multiplied by the conjugate to obtain
>>> (4) from (3).
>>>
>>> A clearer scan may enable me to continue.
>>>
>>> Danke,
>>>
>>
>> (not sure if my response posted as I don't see it on my newsreader,
>> apologies if this reply appears twice)
>>
>> Sure, here's the full page in question:
>>
>> <https://imgur.com/a/as3jfNo>
>>
>> I have a hardcopy from an academic library which is a relatively massive
>> (800+) page tome so difficult to get a good scan of...the only full-text
>> online I can find is on Springerlink (blech) and despite my having an
>> "institutional login" that should grant access to it. it never seems to
>> work with them.
> 
> Your first followup was indeed posted.
> 
> The first three steps from (4) to (5) are easy-peasy:
> 
>      tanh(s) = (e^s - e^-s) / (e^s + e^-s)
>      H(s) = Q(s) / D(s) = (e^s - e^-s) / (e^s + e^-s)
>      (e^s + e^-s)Q(s) = (e^s - e^-s)D(s)
> 
> Control Theory must now be reviewed by me in order to continue.

Thanks, I think I see sorta see how (5) is derived now. I believe they 
mean by their notation tanh(phi(s)) and phi(s) = arctan(Q(s)/D(s)).

The denominator of (4) will be real, and the portions of the numerator 
that are an even function times an even function will be real and the 
parts that are anything else will be imaginary, cuz in e^ix = cos(x) + i 
sin(x) the sin is imaginary and sin is an odd function, when each term 
in the expanded fraction is expressed as a magnitude and phase angle.

Then there's a logarithmic form of the arctangent, arctan(z) = 
-i/2*ln[(1 + iz)/(1 - iz)] and for z(s) = Q(s)/D(s) as decomposed into 
even and odd parts in (4), I think plugging that form into 
tanh(arctan(z)) = (e^z - e^-z) / (e^z + e^-z) should then give (5), 
though I haven't grunged it all out to check.


> # # #
> 
> "Lecture Notes in Control and Information Sciences" seems to be a series
> of books, each about three hundred pages long. Where do you find page
> 808?
> 
> Danke,
> 

This one:

<https://link.springer.com/book/10.1007/BFb0006119?page=6>