[sdiy] 90-degree phase displacement network calculations

[Oren Leavitt]

Pretty cool, David!

I've been looking for a tool to calculate RCs for an n stage all pass 
network.

On 1/5/21 10:46 PM, David G Dixon wrote:
> Well, that was easier than I thought it would be.
> I've now got a function in Excel that predicts the RC factors for the 
> all-pass filters in any 90-degree PDN.  Here is the function:
> PDN90RC(F1, F2, NN, N)
> I've made it so that NN is the number of filter stages in each chain, 
> so half the total.  This way, it confines one to an even number of 
> filter stages.  So, for the classic 15 Hz to 15 kHz unit with 12 
> filter stages, NN = 6, and the function would be:
> PDN90RC(15, 15000, 6, 1) = 0.0301
> If one sets up a column of integers from 1 to 12, and puts the 
> function next to that with those cell values as N, then one gets the 
> following:
> 1 	3.0100E-02
> 2 	8.6072E-03
> 3 	3.8927E-03
> 4 	1.9037E-03
> 5 	9.4806E-04
> 6 	4.7422E-04
> 7 	2.3740E-04
> 8 	1.1875E-04
> 9 	5.9136E-05
> 10 	2.8920E-05
> 11 	1.3080E-05
> 12 	3.7402E-06
>
> I have confirmed that these are exactly the same values generated by 
> the QuadNet program.  The cool thing is that this still works if you 
> exchange the two end frequencies, so:
> PDN90RC(15000, 15, 6, 1) = 0.0301
> Remember that one filter chain takes the odd-numbered RC values, and 
> the other filter chain takes the even-numbered RC values.
> Here is the code:
> ----------------------------------------------------------------------
> Option Base 0
> Function PDN90RC(f1, f2, nn, n)
> Dim k(8) As Double
> Dim kp(8) As Double
> Dim sn(8) As Double
> Pi = 3.14159265359
> f = (f1 * f2) ^ (0.5)
> kp(0) = f1 / f2
> For j = 1 To 8
>   k(j) = (1 - kp(j - 1)) / (1 + kp(j - 1))
>   kp(j) = (1 - k(j) ^ 2) ^ (0.5)
> Next j
> sn(8) = Sin((2 * n - 1) * (Pi / 2) / (4 * nn))
> For j = 7 To 0 Step -1
>   sn(j) = (1 + k(j + 1)) * sn(j + 1) / (1 + k(j + 1) * sn(j + 1) ^ 2)
> Next j
> cn = (1 - sn(0) ^ 2) ^ (0.5)
> pole = kp(0) ^ (-0.5) * cn / sn(0)
> PDN90RC = pole / (2 * Pi * f)
> End Function
> ---------------------------------------------------------------------
> The thing to note about this code is that I discovered that 8 
> successive Landen transformations are enough to get to k = 0 and k' = 
> 1 from any starting k' value, and therefore, sn(u,k) = sin(u).  No 
> matter the value of k' (which is F1/F2), this result is always reached 
> within 8 transformation steps.  Sometimes, of course, it is reached in 
> fewer steps, but I just set the number of transformations to 8.  The 
> variables sn(0) and cn are the Jacobi elliptic sine and cosine, from 
> which the filter poles are calculated directly.
> This was fun.
> Cheers,
> Doc Sketchy
>
> ------------------------------------------------------------------------
> *From:* Synth-diy [mailto:synth-diy-bounces at synth-diy.org] *On Behalf 
> Of *David G Dixon
> *Sent:* Tuesday, January 05, 2021 6:42 PM
> *To:* 'synth-diy at synth-diy.org'
> *Subject:* [sdiy] 90-degree phase displacement network calculations
>
> [*CAUTION:* Non-UBC Email]
> Hello SDIY Team,
> So, ever since I've been building Bode frequency shifters, I've been 
> curious about how to actually design best-case 90-degree phase 
> displacement networks.  Here's a brief history of my experience with this:
> First, I used the numbers in the Electronotes article, EN-168.
> Then I discovered the QuadNet program on the internet, which gives 
> more accurate PDNs than the Electronotes numbers. However, the QuadNet 
> program is a pain to use, and restricts one to certain frequency 
> ranges.  For example, you can't have a low frequency lower than 1 Hz, 
> or a frequency range higher than 4 orders of magnitude.  I want to go 
> lower and broader.
> So, the Electronotes method is an approximation to the problem 
> reported by Weaver in 1954 (which I tracked down).  It's easy to use, 
> but does not give true Chebyshev approximation accuracy.
> I contacted the guy who wrote the QuadNet program to see if he could 
> give me the source of the math he used in the program.  He had 
> forgotten, and didn't have any documentation.
> So, I started looking up the original references for this problem.  
> After much hunting, I found Darlington's paper from 1950 in the Bell 
> Labs technical journal.  It explains the problem, but not in a way in 
> which my tiny brain could make immediate use.
> After a bit more searching, I found a Masters thesis by a guy named 
> Donald Douglas from 1961, and his references included another 
> 1950 paper by a guy named Orchard from a journal called Wireless 
> Engineer.  I finally found a PDF of this article on the web lastnight, 
> and I have now been able to exactly reproduce my designs from QuadNet 
> using the techniques in this paper.  Thankfully, there is a very nice 
> numerical example in the paper which demonstrates the use of the 
> Landen transformation to calculate Jacobi's elliptic sine function 
> from circular sines.  With this, it is trivially easy to find the 
> correct RC factors for the chains of first-degree allpass filters.
> As of now, I can do it very quickly in an Excel spreadsheet, but I'm 
> going to write a function subprogram in Visual Basic that will make 
> the RC factors available as an Excel function -- something like 
> PDN90RC(F1, F2, N, n) where F1 and F2 are the mininum and maximum 
> frequencies, N is the total number of filter stages, and n is the 
> filter stage of interest.
> I'll let you all know when I've got the function subprogram figured 
> out.  Maybe I'll write a little paper about it as well.
> Cheers,
> Doc Sketchy
>
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