[sdiy] 90-degree phase displacement network calculations
David G Dixon
dixon at mail.ubc.ca
Sat Jan 9 07:35:20 CET 2021
Hi Brian,
"Circular" trigonometric functions are just the regular trigonometric
functions. So, "circular" sine is just sine. They trace the dimensions of
triangles around circles.
The Jacobi elliptic functions are generalizations of the trigonometric
functions which trace triangles around ellipses rather than circles. They
have two arguments instead of just one (the angle and the "modulus"), and
they are generally not tabulated but have to be calculated by solving
elliptic integrals parametrically (the answer is the unknown upper limit of
the integral). However, there is this cool concept known as "Landen
transformation" which plays a trick with the integral, and results in a very
simple way to calculate the elliptic functions starting from the circular
functions.
There's a pretty good Wikipedia page about all this stuff.
Some very very smart people figured out how to use all this crap to solve
for the poles of various filter circuits back in the 30s, 40s and 50s, when
radio science was in its heyday. Reading that stuff is very humbling.
Cheers,
Dave
-----Original Message-----
From: Brian Willoughby [mailto:brianw at audiobanshee.com]
Sent: Friday, January 08, 2021 8:41 PM
To: David G Dixon
Cc: *SYNTH DIY
Subject: Re: [sdiy] 90-degree phase displacement network calculations
[CAUTION: Non-UBC Email]
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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