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
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.
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.
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.
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.
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.
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
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