[sdiy] 90-degree phase displacement network calculations

[Ian Fritz]

Well, I can’t get the Orchard paper, so I can’t tell. The Johannesson variant looks like it converges very rapidly, though, at least for his example. So it, too, can give perfect accuracy, in the current sense.

> On Jan 13, 2021, at 2:30 PM, David G Dixon <dixon at mail.ubc.ca> wrote:
> 
> 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
> 
>