[sdiy] 90-degree phase displacement network calculations

[Brian Willoughby]

Hi Dave,

Thank you for your patient summary. Calculating ideal filters is still a mystery that I leave to others.

Brian

On Jan 8, 2021, at 22:35, David G Dixon <dixon at mail.ubc.ca> wrote:
> 
> 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-----
> Sent: Friday, January 08, 2021 8:41 PM
> To: David G Dixon
> Cc: *SYNTH DIY
> 
> 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?