# An s-Plane to z-Plane Mapping Example

While surfing around the Internet recently I encountered the 's-plane to z-plane mapping' diagram shown in Figure 1. At first I thought the diagram was neat because it's a good example of the old English idiom: "A picture is worth a thousand words." However, as I continued to look at Figure 1 I began to detect what I believe are errors in the diagram.

Reader, please take a few moments to see if you detect any errors in Figure 1.

Figure 1: One (incorrect) Internet 's-plane to z-plane
mapping' diagram.

I have drawn what I think is a corrected version of Figure 1. Given the various loci of points in Figure 1's s-plane, my version of a correct 's-plane to z-plane mapping' diagram shown in Figure 2.

Figure 2: Corrected 's-plane to z-plane mapping' diagram.

Hopefully my Figure 2 is worth 1001 words. If there are any errors in that figure I hope a perceptive reader lets me know.

[ - ]
Comment by September 23, 2016
Just eyeballing things -- aren't you both wrong on the constant damping factor line? The angle of departure from z = 1 should be the same as the angle of departure from s = 0. In fact, for any exact mapping -- and therefore any decent approximate mapping -- the behavior of the poles close to z = 1 should just be a scaled and offset version of the behavior of the poles close to s = 0.
[ - ]
Comment by September 23, 2016
You are mistaken. The constant damping factor locus of points (circular light purple curve) on the z-plane is correct. That line starts at a frequency of zero and the frequency increases in the negative-frequency direction. As the frequency goes more and more negative the line's magnitude (e^-0.5) does not change.

Your phrase " The angle of departure from z = 1" makes no sense to me because the constant damping factor locus of points (circular light purple curve) on the z-plane does NOT intersect the z = 1 point.
[ - ]
Comment by September 28, 2016
Tim, I agree with Rick on the constant damping factor line.
[ - ]
Comment by September 28, 2016
Rick, are you sure the mapping the original diagram used was z = e^S and not z = e^S* (conjugate)? You didn't provide a reference to the original diagram so I can't check.

--Randy
[ - ]
Comment by October 2, 2016
Hi Randy. OK, I'm back at my office. I double checked the web page that presented the original mapping diagram. That original mapping diagram was indeed based on the definition z = e^s.
[ - ]
Comment by September 28, 2016
Hi Randy. I'm away from my office for a few days. I can't remember the reference off the top of my head, but I'll get back to you regarding your question.
[ - ]
Comment by March 19, 2021

Hi. Thanks to useful picture. But the bilinear transform is conformal transform, so angle between dark-purple and blue / dark-purple and green must be 45 degrees, as in source picture.

[ - ]
Comment by March 19, 2021

I'm sorry itxs. I don't understand what "angle between dark-purple and blue / dark-purple and green must be 45 degrees, as in source picture" means.

[ - ]
Comment by September 27, 2023

Hi, I am not itxs, but let me clarify a bit:

This means that bilinear transform is a conformal map, so angles between lines should be preserved. Angle formed by the intersecting dark-purple and blue lines in the red point on the right should be equal to 45 degrees, as on the left picture.

One more thing concerns me a bit. Bilinear transform should map lines into lines or circles and circles - into circles or lines. Dark purple line does not meet this requirement.

Hope this helps.

[ - ]
Comment by December 2, 2020

Great diagram. I teach a course in digital control and students tend to struggle with s to z-plane mapping. Picture definitely worth 100 words. Especially like the units of the jw in terms of pi which seems more understandable that units of ws. Thanks for sharing

To post reply to a comment, click on the 'reply' button attached to each comment. To post a new comment (not a reply to a comment) check out the 'Write a Comment' tab at the top of the comments.