[sdiy] 90-degree phase displacement network calculations

David G Dixon dixon at mail.ubc.ca
Wed Jan 13 04:28:02 CET 2021

 -- 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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