HN Reader

I made an interactive implementation here: https://observablehq.com/@mbostock/smith-chart

"Escher's art, Smith Chart and Hyperbolic Geometry"

https://www.researchgate.net/publication/3427377_Escher's_ar...

"A conceptual relation between Circle Limit IV, an artistic creation by M.C. Escher, and Smith Chart, geographical aid for microwave engineering created by P.H. Smith, was established. The basis of Escher's art and Smith chart can both be traced back to invariance of the cross ration of four complex numbers under Möbius transformation on the domain of complex numbers. The Smith chart can be used as an aid for constructing Escher-like drawings that display periodic mosaic patterns and at the same time convey the perception of infinite progression within a unit circle."

This is one of my favorite explanations of how to use a Smith chart in practice: https://www.cypress.com/file/136236/download

Also this one: https://www.ti.com/lit/an/swra046a/swra046a.pdf

This animation is helpful:

https://en.wikipedia.org/wiki/Smith_chart#/media/File:Animat...

I've used Smith charts many times on my NanoVNA analysing antennas, but despite being mathematically inclined I never thought of the complex plane mapping involved

f(z) = (z − 1)/(z + 1)

The Smith chart is useful to electronic engineers because a given VSWR (the thing you try to minimise to get a good antenna) becomes a circle about the center of the Smith chart

VSWR= (1 + | Γ |)/(1 - | Γ |)

So to make your antenna better, get the plot closer to the center. Whether it is above the line or below tells you whether the antenna is inductive or capacitive and hence which kind of loading to add.

I loved smith charts during my microwave classes in college. I've always felt like there should be a fun game based on the mechanics of using a smith chart.

> The Smith chart from electrical engineering is the image of a Cartesian grid under the function f(z) = (z − 1)/(z + 1). More specifically, it’s the image of a grid in the right half-plane.

Well, I'm already confused. The domain is complex numbers where the real part is >= 0? I think it would be helpful to make that clear from the get-go. When I see "Cartesian grid", I think R^2, not C.

Btw, as all Möbius transforms, a Smith chart can be understood by looking at the complex plane as a sphere:

https://www.youtube.com/watch?v=0z1fIsUNhO4

I have PTSD from learning Smith charts in school.

My prof even said to me "If you'd pay attention, you'd understand this!". I was the only one who was brave enough to ask questions about it!

Smith charts look really cool, and they can be really useful for modeling, but for me I can't see the phasors when I use a Smith chart. Myself and all the EEs I work with use bode plots to see mag and phase response across frequency. I know some sick f EEs that just want raw IQ pairs. But Smith charts look sci fi so they're good too.

Is this related to Cardioid patterns for directional microphones?

Why would you use one?

Smith Chart 101: https://www.youtube.com/watch?v=vDU5XnvZXwc

These are wormhole graphs, right?

Terrifying. My nightmares will return.