# [sdiy] 90-degree phase displacement network calculations

David G Dixon dixon at mail.ubc.ca
Mon Jan 11 21:42:04 CET 2021

```Hello Ian,

Cheers
Dave

-----Original Message-----
From: Ian Fritz [mailto:ijfritz at comcast.net]
Sent: Monday, January 11, 2021 6:44 AM
To: David G Dixon; 'Bernard Arthur Hutchins, Jr'; synth-diy at synth-diy.org
Cc: 'Brian Willoughby'
Subject: Re: [sdiy] 90-degree phase displacement network calculations

[CAUTION: Non-UBC Email]

That looks not to be true. The difference between two successive k'(i)
values clearly can not be zero. The process is a (rapidly) converging
iterative one.

In case you can't see this, the proof is trivial:
Suppose k'(i) = k'(i-1)
Then from the second equation, k(i) = k(i-1) Now the first equation yields 0
= k(i)-k(i-1) = [1-k'(i-1)]/[1+k'(i-1)] -
[1-k'(i-2)]/[1+k'(i-2)]
This can be generally true only if k'(i-2) = k'(i-1) So by induction, all
the k'(i) values are the same.

A sequence either iterates or it doesn't -- it can't just drop dead in the
middle of the street.

Ian
(math minor, including some pretty tough analysis courses)

On 1/11/2021 2:23 AM, David G Dixon wrote:

> ........  There are no "approximate" answers, and this problem is not
> one where one gets closer and closer to the true solution with each
> step.  That would be an iterative solution, and as I've said ad
> nauseum, this is not that problem.

```