Really cool! BTW since you're here, have you checked https://github.com/malloydata/malloy/ and their visualizations? it seems it could be a nice fit for ObservableHQ
"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."
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.
Not knowing enough about Smith Charts, do the intermediary frames have any meaning in terms of actual practical problems or calculations?
It seems like the chart itself would be a member of an entire family of chart types that would go through the entire 3D rotation with several versions that were "nearly tangent circles" and eventually concentric circles at the center of the graph, along with some of the intermediary frames shown in the animation.
The intermediary frames have an interpretation as being congruent to a scaling. Near the outset of the animation when the imaginary axis slightly bends, it has already become a circle. So it is congruent to a large Smith chart.
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.
I felt the same way. At some point shortly after graduation, I found a heavy plastic Smith chart that was designed to be written on with a grease pencil, and erased with a cloth when you're done.
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.
> 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.
So you can continue it beyond that circle if you like, it just happens to be the case that the thing electrical engineers and others dealing with waveguides are plotting in the source space, is impedance. (Well, a ratio of impedances—a load impedance divided by a transmission line impedance.) The real part of impedance is resistance, and negative resistance is very uncommon. The area outside the unit circle so mapped, I think also corresponds to reflected amplitude ≤ transmitted amplitude, with the center of the diagram being a perfectly matched impedance, and no reflection.
I switched from EE to CS (well, "Computer Engineering" technically) in the late 90s. Not specifically due to Smith charts, but that's relatable. For me it was just realizing that I was procrastinating on doing my EE problem sets, which just started to seem like endless grinding of differential equations, by playing around with whatever we were doing in the couple CS classes I had. I wouldn't say I've made "a large fortune" in software, but it's kept me gainfully employed for a few decades so I think it worked out.
No. The RF equivalent of that would be an antenna pattern. The smith chart is a particular mapping of complex impedance that allows for fairly straightforward calculation and visualization of things like conjugate matches. It's related to the impedance plots for speakers, but audio engineers generally don't use it. RF engineers will plot impedance vs frequency on it though. If you come across curlicue plots on it, those are typically frequency sweeps i.e. they plot impedance vs frequency. You might have to look for the frequency legend or endpoints.
I remember using these for impedance matching back when I was in college. Basically when you connect two transmission lines (like coax cables), you need to match their impedances so the signal does not bounce back. (Ik this is a gross oversimplification but yeah)
I made an interactive implementation here: https://observablehq.com/@mbostock/smith-chart
Really cool! BTW since you're here, have you checked https://github.com/malloydata/malloy/ and their visualizations? it seems it could be a nice fit for ObservableHQ
that's beautiful!
Really really nice. Thanks for this!
"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
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.
> f(z) = (z − 1)/(z + 1)
Also known as the Bilinear Transform https://en.wikipedia.org/wiki/Bilinear_transform
Used for mapping between s-plane and z-plane when discretising using the trapezoidal rule.
This animation is helpful:
https://en.wikipedia.org/wiki/Smith_chart#/media/File:Animat...
Not knowing enough about Smith Charts, do the intermediary frames have any meaning in terms of actual practical problems or calculations?
It seems like the chart itself would be a member of an entire family of chart types that would go through the entire 3D rotation with several versions that were "nearly tangent circles" and eventually concentric circles at the center of the graph, along with some of the intermediary frames shown in the animation.
The intermediary frames have an interpretation as being congruent to a scaling. Near the outset of the animation when the imaginary axis slightly bends, it has already become a circle. So it is congruent to a large Smith chart.
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.
This software is so good it is as enjoyable as any game, it taught me to love Smith Charts
https://www.ae6ty.com/smith_charts/
Oh, found a picture of one of these plastic ones flikr:
https://www.flickr.com/photos/jonsphotos/218609214/in/album-...
I’m just amazed there isn’t a nearly impenetrable Greg Egan book based on them.
If you liked the Smith Chart, you'd love using the Spirule to solve root locus problems.
https://en.wikipedia.org/wiki/Root_locus_analysis
I felt the same way. At some point shortly after graduation, I found a heavy plastic Smith chart that was designed to be written on with a grease pencil, and erased with a cloth when you're done.
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
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.
> 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.
So you can continue it beyond that circle if you like, it just happens to be the case that the thing electrical engineers and others dealing with waveguides are plotting in the source space, is impedance. (Well, a ratio of impedances—a load impedance divided by a transmission line impedance.) The real part of impedance is resistance, and negative resistance is very uncommon. The area outside the unit circle so mapped, I think also corresponds to reflected amplitude ≤ transmitted amplitude, with the center of the diagram being a perfectly matched impedance, and no reflection.
Smith Chart 101: https://www.youtube.com/watch?v=vDU5XnvZXwc
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!
Heh, this reminds me, don't worry:
One of my extremely intelligent roommates in the 80s switched from EE to CS, seemingly due to Smith charts and Electromagnetics coursework.
He went on to make a large fortune in software.
I switched from EE to CS (well, "Computer Engineering" technically) in the late 90s. Not specifically due to Smith charts, but that's relatable. For me it was just realizing that I was procrastinating on doing my EE problem sets, which just started to seem like endless grinding of differential equations, by playing around with whatever we were doing in the couple CS classes I had. I wouldn't say I've made "a large fortune" in software, but it's kept me gainfully employed for a few decades so I think it worked out.
Sounds like it was the right decision then.
Is this related to Cardioid patterns for directional microphones?
No. The RF equivalent of that would be an antenna pattern. The smith chart is a particular mapping of complex impedance that allows for fairly straightforward calculation and visualization of things like conjugate matches. It's related to the impedance plots for speakers, but audio engineers generally don't use it. RF engineers will plot impedance vs frequency on it though. If you come across curlicue plots on it, those are typically frequency sweeps i.e. they plot impedance vs frequency. You might have to look for the frequency legend or endpoints.
No. Sorry, that is not a helpful comparison.
Why would you use one?
I remember using these for impedance matching back when I was in college. Basically when you connect two transmission lines (like coax cables), you need to match their impedances so the signal does not bounce back. (Ik this is a gross oversimplification but yeah)
Terrifying. My nightmares will return.
These are wormhole graphs, right?
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