# [sdiy] 90-degree phase displacement network calculations

Brian Willoughby brianw at audiobanshee.com
Sat Jan 9 05:41:08 CET 2021

```On Jan 8, 2021, at 12:48, David G Dixon <dixon at mail.ubc.ca> wrote:
> The Orchard paper has a numerical example (for a 4-stage network) and this is what allowed me to figure out how to do the math.  The math ultimately is very simple.  If the total number of stages is 2N (I'm restricting it to even-numbered stages), then the pole values are given as:
>
> (1/k')^(0.5) * sn(2n*pi/4N,k) / cn(2n*pi/4N,k)
>
> where n goes from 1 to N.  k' = f1/f2 where f1 and f2 are the endpoint frequencies of the desired bandwidth.  sn and cn are Jacobi's elliptic sine and cosine.  k (the modulus of the Jacobi functions) = (1 - k'^2)^(0.5).  To calculate these Jacobi functions, one uses the Landen transformation, which allows the reduction of the k and k' values using a recursive formula until k = 0 and k' = 1.  This formula is simply that the next value of k in the table = (1 - k')/(1 + k').  One then calculates the new value of k' = (1 - k^2)^(0.5) and continues until k = 0 and k' = 1 (which I have confirmed takes no more than 8 steps regardless of the initial value of k').  Once k' = 1, the Jacobi sn function is equivalent to the circular sine function.

I have never heard the terms, elliptic sine, or circular sine. Is there a simple definition of these?

I ask because if you sum the odd harmonics of a sine wave fundamental, and weight each harmonic by 1/(N*N), you end up with a waveform that looks like a circular sine (to me).

... and, if you use cosine instead, or merely invert the polarity of every other harmonic, then the same weighting produces a triangle wave. The triangle wave sounds exactly the same as this "circular" wave, since we're not sensitive to polarity. An advantage is that the "circular" wave is louder than a triangle wave for the sample peak-to-peak amplitude. It's more difficult to generate in the analog domain, but simple in the digital domain.

Sorry for the off-topic distraction. I'm just curious about the terminology.

Brian

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