[sdiy] 90-degree phase displacement network calculations

Ian Fritz ijfritz at comcast.net
Wed Jan 13 21:24:11 CET 2021

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.


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


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.


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