[sdiy] 90-degree phase displacement network calculations

[David G Dixon]

Interesting.  However, the Landen transformation approach is actually easier
to implement in code, and gives perfect accuracy, whereas Johannesson's
approach is approximate.

In 1950, the repeating algorithm would have been very tedious indeed, but
today it is trivial.

The Weaver method is a similar approximation, which gives less optimal but
still useful results, but again, Orchard's approach is actually easier to
implement in code.
 

-----Original Message-----
From: Ian Fritz [mailto:ijfritz at comcast.net] 
Sent: Wednesday, January 13, 2021 12:24 PM
To: David G Dixon; 'Bernard Arthur Hutchins, Jr'; synth-diy at synth-diy.org
Subject: Re: [sdiy] 90-degree phase displacement network calculations

[CAUTION: Non-UBC Email]

In a note re the Orchard paper, Johannesson presented a modification that
does not require the "repeated Landen transformations". It uses just a few
terms from expansions.  He says his numerical results for the Orchard
example agree with the original.

https://worldradiohistory.com/UK/Experimental-Wireless/50s/Wireless-Engineer
-1950-08-09.pdf

Thanks to Bernie for including this reference in his early article.

Ian


On 1/12/2021 8:28 PM, David G Dixon wrote:
>   -- I understand what you did and for the third time state that there 
> is nothing wrong with what you did.
> 
> I already knew that, Bernie.  I wasn't seeking your approval.  I was 
> just presenting a technique for everyone to use as they wish.
> 
>   -- But you have avoided my repeated questions about what the error 
> plot looks like if you do fewer iterations. [ Neither have you 
> commented your program or (despite saying there were several errors 
> along the way) provided a full corrected copy. ]
> 
> OK, I'll say it one more time: There are no iterations.  If one does 
> not take the necessary number of steps to complete the Landen 
> transformation, then one gets inaccurate numbers for the elliptic sine 
> and cosine.  I see no purpose to calculating anything if the elliptic 
> functions are not accurately calculated.  Once the necessary elliptic 
> functions are calculated, then the pole values are calculated simply 
> from those.
> 
> To do what you request would be like calculating a trigonometric 
> function with sines and cosines, but instead of actually using those 
> functions, using series expansions truncated at the third step.  Why 
> would anyone do that?
> 
> Orchard presents an equation to calculate the maximum error in the 
> phase angle directly.  However, it does not seem to give the correct
answer.
> For phase angle, I am calculating it directly from the RC factors of 
> the filter, and summing up the total phase angle for all stages in 
> each chain, then subtracting one phase angle from the other.  I have 
> benchmarked my results against the QuadNet program, and the two 
> results are identical.  The equation in Orchard dramatically 
> underestimates the maximum error. For example, a 16-stage PDN with a 
> bandwidth of 4 decades will generate a maximum phase error of 0.133 
> degrees.  The Orchard equation gives 0.066 degrees.  With a bandwidth 
> of 5 decades the maximum error increases to about 0.5 degrees.  The 
> Orchard equation gives 0.116 degrees.  So, that's a fail.  It bears 
> noting that the equation given in Darlington for this purpose is 
> different, so the old guard were not all on the same page, evidently.
> 

--
ijfritz.byethost4.com